ࡱ> 1<>56789:;NPiqsuz | ~ B DFHJLNPRT)` {bjbj 4^T&8&BjB,>nE"EEEkFiHIX  hrJkFkFJJEE$MMMJEE MJ MME ŋE6 wNlIKQ:d:0j:Ktŋŋ$mI>I,MI$ImImImIgMXmImImIjJJJJd*k* 10.1 Integration by Substitution Each basic rule of integration that you have studied so far was derived from a corresponding differentiation rule. Even though you have learned all the necessary tools for differentiating exponential, logarithmic, trigonometric, and algebraic functions, your set of tools for integrating these functions is not yet complete. In this chapter we will explore different ways of integrating functions and develop several integration techniques that will greatly expand the set of integrals to which the basic integration formulas can be applied. Before we do that, let us review the basic integration formulas that you are already familiar with from previous chapters. 1. The Power Rule  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28n%20%5Cneq%20-1%29" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20x%5En%20dx%20%3D%20%5Cfrac%7Bx%5En%20%2B%201%7D%20%7Bn%20%2B%201%7D%20%2B%20C." \* MERGEFORMATINET  2. The General Power Rule  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28n%20%5Cneq%20-1%29" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20u%5En%20%5Cfrac%7Bdu%7D%20%7Bdx%7D%20dx%20%3D%20%5Cint%20u%5En%20du%20%3D%20%5Cfrac%7Bu%5E%7Bn%20%2B%201%7D%7D%20%7Bn%20%2B%201%7D%20%2B%20C." \* MERGEFORMATINET  3. The Simple Exponential Rule:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20e%5Ex%20dx%20%3D%20e%5Ex%20%2B%20C." \* MERGEFORMATINET  4. The General Exponential Rule:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20e%5Eu%20%5Cfrac%7Bdu%7D%20%7Bdx%7D%20dx%20%3D%20%5Cint%20e%5Eu%20du%20%3D%20e%5Eu%20%2B%20C." \* MERGEFORMATINET  5. The Simple Log Rule:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7B1%7D%20%7Bx%7D%20dx%20%3D%20%5Cln%20%7Cx%7C%20%2B%20C." \* MERGEFORMATINET  6. The General Log Rule:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7Bdu/dx%7D%20%7Bu%7D%20dx%20%3D%20%5Cint%20%5Cfrac%7B1%7D%20%7Bu%7D%20du%20%3D%20%5Cln%20%7Cu%7C%2B%20C." \* MERGEFORMATINET  It is important that you remember the above rules because we will be using them extensively to solve more complicated integration problems. The skill that you need to develop is to determine which of these basic rules is needed to solve an integration problem. Learning Objectives A student will be able to: Compute by hand the integrals of a wide variety of functions by using the technique of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution. Apply the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution technique to definite integrals. Apply the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution technique to trig functions. Probably one of the most powerful techniques of integration is integration by substitution. In this technique, you choose part of the integrand to be equal to a variable we will call  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and then write the entire integrand in terms of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u." \* MERGEFORMATINET The difficulty of the technique is deciding which term in the integrand will be best for substitution by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u." \* MERGEFORMATINET However, with practice, you will develop a skill for choosing the right term. Recall from Chapter 2 that if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET is a differentiable function of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET is a real number and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%5Cneq%20-1%2C" \* MERGEFORMATINET then the Chain Rule tells us that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7Bd%7D%20%7Bdx%7D%20%5Bu%5En%5D%20%3D%20nu%5E%7Bn%20-%201%7D%20%5Cfrac%7Bdu%7D%20%7Bdx%7D." \* MERGEFORMATINET  The reverse of this formula is the integration formula,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20u%5En%20du%20%3D%20%5Cfrac%7Bu%5E%7Bn%20%2B%201%7D%7D%20%7Bn%20%2B%201%7D%20%2B%20C%2C%20n%20%5Cne%20-1." \* MERGEFORMATINET  Sometimes it is not easy to integrate directly. For example, look at this integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%285x%20-%202%29%5E2%20dx." \* MERGEFORMATINET  One way to integrate is to first expand the integrand and then integrate term by term.  That is easy enough. However, what if the integral was  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%285x%20-%202%29%5E%7B15%7D%20dx%3F" \* MERGEFORMATINET  Would you still expand the integrand and then integrate term by term? That would be impractical and time-consuming. A better way of doing this is to change the variables. Changing variables can often turn a difficult integral, such as the one above, into one that is easy to integrate. The method of doing this is called integration by substitution, or for short, the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET -substitution method. The examples below will show you how the method is used. Example 1: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%28x%20%2B%201%29%5E5%20dx." \* MERGEFORMATINET  Solution: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x%20%2B%201." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20d%28x%20%2B%201%29%20%3D%201dx%20%3D%20dx." \* MERGEFORMATINET Substituting for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du" \* MERGEFORMATINET we get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%28x%20%2B%201%29%5E5%20dx%20%3D%20%5Cint%20u%5E5%20du." \* MERGEFORMATINET  Integrating using the power rule,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%3D%20%5Cfrac%7Bu%5E6%7D%20%7B6%7D%20%2B%20C." \* MERGEFORMATINET  Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x%20%2B%201%2C" \* MERGEFORMATINET substituting back,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%3D%20%5Cfrac%7B%28x%20%2B%201%29%5E6%7D%20%7B6%7D%20%2B%20C." \* MERGEFORMATINET  Example 2: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csqrt%7B4x%20%2B%203%7Ddx." \* MERGEFORMATINET  Solution: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%204x%20%2B%203." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%204dx." \* MERGEFORMATINET Solving for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=dx%20%26%20%3D%20du/4." \* MERGEFORMATINET  Substituting,  Simplifying,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%26%20%3D%20%5Cfrac%7B1%7D%20%7B6%7D%20u%5E%7B3/2%7D%20%2B%20C%5C%5C%0A%26%20%3D%20%5Cfrac%7B1%7D%20%7B6%7D%20%284x%20%2B%203%29%5E%7B3/2%7D%20%2B%20C." \* MERGEFORMATINET  Trigonometric Integrands We can apply the change of variable technique to trigonometric functions as long as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET is a differentiable function of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET Before we show how, recall the basic trigonometric integrals:  Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ccos%283x%20%2B%202%29dx." \* MERGEFORMATINET  Solution: The argument of the cosine function is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3x%20%2B%202." \* MERGEFORMATINET So we let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%203x%20%2B%202." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%203dx%2C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%20%3D%20du/3." \* MERGEFORMATINET  Substituting,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ccos%283x%20%2B%202%29dx%20%26%20%3D%20%5Cint%20%5Ccos%20u%20%5Ccdot%20%5Cfrac%7B1%7D%20%7B3%7D%20dx%5C%5C%0A%26%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20%5Cint%20%5Ccos%20u%20dx." \* MERGEFORMATINET  Integrating,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%26%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20%5Csin%20u%20%2B%20C%5C%5C%0A%26%20%3D%20%5Cfrac%7B1%7D%20%7B3%7D%20%5Csin%283x%20%2B%202%29%20%2B%20C." \* MERGEFORMATINET  Example 4: This example requires us to use trigonometric identities before we substitute. Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7B1%7D%20%7Bcos%5E2%203x%7D%20dx." \* MERGEFORMATINET  Solution: Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csec%203x%20%3D%20%5Cfrac%7B1%7D%20%7B%5Ccos%203x%7D" \* MERGEFORMATINET , the integral becomes  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7B1%7D%20%7Bcos%5E2%203x%7D%20dx%20%3D%20%5Cint%20%5Csec%5E2%203x%20dx." \* MERGEFORMATINET  Substituting for the argument of the secant,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%203x%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%203dx%2C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%20%3D%20du/3." \* MERGEFORMATINET Thus our integral becomes,  Some integrations of trigonometric functions involve the logarithmic functions as a solution, as shown in the following example. Example 5: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ctan%20x%20dx" \* MERGEFORMATINET . Solution: As you may have guessed, this is not a straightforward integration. We need to make use of trigonometric identities to simplify it. Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctan%20x%20%3D%20%5Csin%20x/%5Ccos%20x%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%20x%20dx%20%3D%20%5Cint%20%5Cfrac%7B%5Csin%20x%7D%20%7B%5Ccos%20x%7D%20dx." \* MERGEFORMATINET  Now make a change of variable  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ccos%20x." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20-%5Csin%20x%20dx%2C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%20%3D%20-du/%5Csin%20x." \* MERGEFORMATINET Substituting,  This integral should look obvious to you. The integrand is the derivative of the natural logarithm  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%20u." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%26%20%3D%20-%5Cln%7Cu%7C%20%2B%20C%5C%5C%0A%26%20%3D%20-%5Cln%7C%5Ccos%20x%7C%20%2B%20C." \* MERGEFORMATINET  Another way of writing it, since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-%5Cln%20%7Cu%7C%20%3D%20%5Cln%20%5Cfrac%7B1%7D%20%7B%7Cu%7C%7D" \* MERGEFORMATINET , is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%26%20%3D%20%5Cln%20%5Cleft%20%7C%5Cfrac%7B1%7D%20%7B%5Ccos%20x%7D%5Cright%20%7C%20%2B%20C%5C%5C%0A%26%20%3D%20%5Cln%20%7C%5Csec%20x%7C%20%2B%20C." \* MERGEFORMATINET  Using Substitution on Definite Integrals Example 6: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_1%5E3%20%5Cfrac%7Bx%7D%20%7B%5Csqrt%7B2x%20-%201%7D%7D%20dx." \* MERGEFORMATINET  Solution: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%202x%20-%201." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%202dx%2C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%20%3D%20du/2." \* MERGEFORMATINET Before we substitute, we need to determine the new limits of integration in terms of the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET variable. To do so, we simply substitute the limits of integration into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%202x%20-%201" \* MERGEFORMATINET : Lower limit: For  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201%2C%20u%20%3D%202%281%29%20-%201%20%3D%201." \* MERGEFORMATINET  Upper limit: For  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203%2C%20u%20%3D%202%283%29%20-%201%20%3D%205." \* MERGEFORMATINET  We now substitute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and the associated limits into the integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_1%5E5%20%5Cfrac%7Bx%7D%20%7B%5Csqrt%7Bu%7D%7D%20%5Cfrac%7Bdu%7D%20%7B2%7D." \* MERGEFORMATINET  As you may notice, the variable  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET is still hanging there. To write it in terms of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%2C" \* MERGEFORMATINET since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%202x%20-%201%2C" \* MERGEFORMATINET solving for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%2C" \* MERGEFORMATINET we get,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%28u%20%2B%201%29/%282%29." \* MERGEFORMATINET Substituting back into the integral,  Applying the Fundamental Theorem of Calculus by inserting the limits of integration and calculating,  Calculating and simplifying, we get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%3D%20%5Cfrac%7B4%5Csqrt%7B5%7D%20-%20%7B2%7D%7D%20%7B3%7D." \* MERGEFORMATINET  We could have chosen  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Csqrt%7B2x%20-%201%7D" \* MERGEFORMATINET instead. You may want to try to solve the integral with this substitution. It might be easier and less tedious. Example 7: Lets try the substitution method of definite integrals with a trigonometric integrand. Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E%7B%5Cpi/4%7D%20%5Ctan%20x%20%5Csec%5E2%20x%20dx" \* MERGEFORMATINET . Solution: Try  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ctan%20x." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20%5Csec%5E2%20x%20dx%2C%20dx%20%3D%20du/%20%5Csec%5E2%20x." \* MERGEFORMATINET  Lower limit: For  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C%20u%20%3D%20%5Ctan%200%20%3D%200." \* MERGEFORMATINET  Upper limit: For  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpi/4%2C%20u%20%3D%20%5Ctan%20%5Cfrac%7B%5Cpi%7D%20%7B4%7D%20%3D%201." \* MERGEFORMATINET  Thus  Multimedia Links For video presentations on integration by substitution (17.0), see  HYPERLINK "http://www.youtube.com/watch?v=VfvJ5C9FjcA" Math Video Tutorials by James Sousa, Integration by Substitution, Part 1 of 2 (9:42) INCLUDEPICTURE "http://i3.ytimg.com/vi/VfvJ5C9FjcA/hqdefault.jpg" \* MERGEFORMATINET  and  HYPERLINK "http://www.youtube.com/watch?v=pmekyFVIorE" Math Video Tutorials by James Sousa, Integration by Substitution, Part 2 of 2 (8:17) INCLUDEPICTURE "http://i1.ytimg.com/vi/pmekyFVIorE/hqdefault.jpg" \* MERGEFORMATINET . Review Questions In the following exercises, evaluate the integrals.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B3%7D%20%7B%28x%20-%208%29%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csqrt%7B2%20%2B%20x%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B1%7D%20%7B%5Csqrt%7B2%20%2B%20x%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7Bx%5E2%7D%20%7Bx%20%2B%201%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7Be%5E%7B-x%7D%7D%20%7Be%5E%7B-x%7D%20%2B%202%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B3%5Csqrt%7Bt%7D%20%2B%205%7D%20%7Bt%7D%20dt" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B2%7D%20%7B%5Csqrt%7B3x%20-%201%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%20x%20%5Ccos%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ccos%20x%20%5Csqrt%7B1%20-%20%5Ccos%5E2%20x%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%5E5%20x%20%5Ccos%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5E3%20%5Ccos%204x%5E4%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csec%5E2%282x%20%2B%204%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E2%20xe%5E%7Bx%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E%7B%5Csqrt%7B%5Cpi%7D%7D%20x%20%5Csin%20x%5E2%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E1%20x%28x%20%2B%205%29%5E4%20dx" \* MERGEFORMATINET  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-3%7D%20%7Bx%20-%208%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2%7D%20%7B3%7D%20%282%20%2B%20x%29%5E%7B3/2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2%5Csqrt%7B2%20%2B%20x%7D%20%2B%20C" \* MERGEFORMATINET   EMBED Equation.DSMT4   EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=6%5Csqrt%7Bt%7D%20%2B%205%20%5Cln%20%7Ct%7C%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B4%7D%20%7B3%7D%20%5Csqrt%7B3x%20-%201%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D%20%5Csin%5E2%20x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-1%7D%20%7B2%7D%20%5Ccos%5E2%20x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B6%7D%20%5Csin%5E6%20x%20%2B%20C%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B16%7D%20%5Csin%204x%5E4%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D%20%5Ctan%282x%20%2B%204%29%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D%20%28e%5E4%20-%201%29" \* MERGEFORMATINET  1  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=520%5Cfrac%7B5%7D%7B6%7D" \* MERGEFORMATINET  Integration by Substitution HW Integrate! 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  Answers: 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  10.2 Integration By Parts Learning Objectives A student will be able to: Compute by hand the integrals of a wide variety of functions by using technique of Integration by Parts. Combine this technique with the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution method to solve integrals. Learn to tabulate the technique when it is repeated. In this section we will study a technique of integration that involves the product of algebraic and exponential or logarithmic functions, such as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20x%20%5Cln%20x%20dx" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20x%20e%5Ex%20dx." \* MERGEFORMATINET  Integration by parts is based on the product rule of differentiation that you have already studied:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7Bd%7D%20%7Bdx%7D%20%5Buv%5D%20%3D%20u%20%5Cfrac%7Bdv%7D%20%7Bdx%7D%20%2B%20v%20%5Cfrac%7Bdu%7D%20%7Bdx%7D." \* MERGEFORMATINET  If we integrate each side,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=uv%20%26%20%3D%20%5Cint%20u%20%5Cfrac%7Bdv%7D%20%7Bdx%7D%20dx%20%2B%20%5Cint%20v%20%5Cfrac%7Bdu%7D%20%7Bdx%7D%20dx%5C%5C%0A%26%20%3D%20%5Cint%20u%20dv%20%2B%20%5Cint%20v%20du." \* MERGEFORMATINET  Solving for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20u%20dv%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20u%20dv%20%26%20%3D%20u%20v%20-%20%5Cint%20vdu." \* MERGEFORMATINET  This is the formula for integration by parts. With the proper choice of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%2C" \* MERGEFORMATINET the second integral may be easier to integrate. The following examples will show you how to properly choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv." \* MERGEFORMATINET  Example 1: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%20%5Csin%20x%20dx" \* MERGEFORMATINET . Solution: We use the formula  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20u%20dv%20%3D%20uv%20-%20%5Cint%20v%20du" \* MERGEFORMATINET . Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=u%20%3Dx" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=dv%20%3D%20%5Csin%20x%20dx." \* MERGEFORMATINET  To complete the formula, we take the differential of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET  and the simplest antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20%5Csin%20x%20dx." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=du%20%26%20%3D%20dx%5C%5C%0Av%20%26%20%3D%20-%5Ccos%20x." \* MERGEFORMATINET  The formula becomes  A Guide to Integration by Parts Which choices of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET lead to a successful evaluation of the original integral? In general, choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET to be something that simplifies when differentiated, and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET to be something that remains manageable when integrated. Looking at the example that we have just done, we chose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20%5Csin%20x%20dx." \* MERGEFORMATINET That led to a successful evaluation of our integral. However, lets assume that we made the following choice,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=u%20%26%20%3D%20%5Csin%20x%5C%5C%0Adv%20%26%20%3D%20x%20dx." \* MERGEFORMATINET  Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=du%20%26%20%3D%20%5Ccos%20x%20dx%5C%5C%0Av%20%26%20%3D%20x%5E2/2." \* MERGEFORMATINET  Substituting back into the formula to integrate, we get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20u%20dv%20%26%20%3D%20u%20v%20-%20%5Cint%20v%20du%5C%5C%0A%26%20%3D%20%5Csin%20x%20%5Cfrac%7Bx%5E2%7D%20%7B2%7D%20-%20%5Cint%20%5Cfrac%7Bx%5E2%7D%20%7B2%7D%20%5Ccos%20x%20dx" \* MERGEFORMATINET  As you can see, this integral is worse than what we started with! This tells us that we have made the wrong choice and we must change (in this case switch) our choices of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv." \* MERGEFORMATINET  Remember, the goal of the integration by parts is to start with an integral in the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20u%20dv" \* MERGEFORMATINET that is hard to integrate directly and change it to an integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20v%20du" \* MERGEFORMATINET that looks easier to evaluate. However, here is a general guide that you may find helpful: Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET to be the more complicated portion of the integrand that fits a basic integration formula. Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET to be the remaining term in the integrand. Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET to be the portion of the integrand whose derivative is simpler than  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u." \* MERGEFORMATINET Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET to be the remaining term. Example 2: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%20e%5Ex%20dx." \* MERGEFORMATINET  Solution: Again, we use the formula  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20u%20dv%20%3D%20uv%20-%20%5Cint%20v%20du" \* MERGEFORMATINET . Let us choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=u%20%3D%20x" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=dv%20%3D%20e%5Exdx." \* MERGEFORMATINET  We take the differential of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and the simplest antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20e%5Exdx" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=du%20%26%20%3D%20dx%5C%5C%0Av%20%26%20%3D%20e%5Ex." \* MERGEFORMATINET  Substituting back into the formula,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20u%20dv%20%26%20%3D%20uv%20-%20%5Cint%20v%20du%5C%5C%0A%26%20%3D%20x%20e%5Ex%20-%20%5Cint%20e%5Ex%20dx." \* MERGEFORMATINET  We have made the right choice because, as you can see, the new integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20v%20du%20%3D%20%5Cint%20e%5Ex%20dx" \* MERGEFORMATINET is definitely simpler than our original integral. Integrating, we finally obtain our solution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20x%20e%5Ex%20dx%20%3D%20x%20e%5Ex%20-%20e%5Ex%20%2B%20C." \* MERGEFORMATINET  Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cln%20x%20dx" \* MERGEFORMATINET . Solution: Here, we only have one term,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%20x." \* MERGEFORMATINET We can always assume that this term is multiplied by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cln%20x%201%20dx." \* MERGEFORMATINET  So let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Cln%20x%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%201dx." \* MERGEFORMATINET Thus  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%201/x%20dx" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=v%20%3D%20x." \* MERGEFORMATINET Substituting,  Repeated Use of Integration by Parts Oftentimes we use integration by parts more than once to evaluate the integral, as the example below shows. Example 4: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5E2%20e%5Ex%20dx" \* MERGEFORMATINET . Solution: With  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x%5E2%2C%20dv%20%3D%20e%5Ex%20dx%2C%20du%20%3D%202xdx%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=v%20%3D%20e%5Ex%2C" \* MERGEFORMATINET our integral becomes  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20x%5E2%20e%5Ex%20dx%20%3D%20x%5E2%20e%5Ex%20-%202%20%5Cint%20x%20e%5Ex%20dx." \* MERGEFORMATINET  As you can see, the integral has become less complicated than the original,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E2%20e%5Ex%20%5Crightarrow%20x%20e%5Ex" \* MERGEFORMATINET . This tells us that we have made the right choice. However, to evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20xe%5Ex%20dx" \* MERGEFORMATINET we still need to integrate by parts with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20e%5Ex%20dx." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20dx" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=v%20%3D%20e%5Ex%2C" \* MERGEFORMATINET and  Actually, the method that we have just used works for any integral that has the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5En%20e%5Ex%20dx" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET is a positive integer. The following section illustrates a systematic way of solving repeated integrations by parts. Tabular Integration by Parts Sometimes, we need to integrate by parts several times. This leads to cumbersome calculations. In situations like these it is best to organize our calculations to save us a great deal of tedious work and to avoid making unpredictable mistakes. The example below illustrates the method of tabular integration. Example 5: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5E2%20%5Csin%203x%20dx" \* MERGEFORMATINET . Solution: Begin as usual by letting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x%5E2" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20%5Csin%203x%20dx." \* MERGEFORMATINET Next, create a table that consists of three columns, as shown below: Alternate signs  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET and its derivatives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET and its antiderivatives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%2B" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E2%20%5Csearrow" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csin%203x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2x%20%5Csearrow" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-1%7D%20%7B3%7D%20%5Ccos%203x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%2B" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2%20%5Csearrow" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-1%7D%20%7B9%7D%20%5Csin%203x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B27%7D%20%5Ccos%203x" \* MERGEFORMATINET To find the solution for the integral, pick the sign from the first row  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28%2B%29%2C" \* MERGEFORMATINET multiply it by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET of the first row  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28x%5E2%29" \* MERGEFORMATINET and then multiply by the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET of the second row,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-1/3%20%5Ccos%203x" \* MERGEFORMATINET (watch the direction of the arrows.) This is the first term in the solution. Do the same thing to obtain the second term: Pick the sign from the second row, multiply it by the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u" \* MERGEFORMATINET of the same row and then follow the arrow to multiply the product by the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv" \* MERGEFORMATINET in the third row. Eventually we obtain the solution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20x%5E2%20%5Csin%203x%20dx%20%3D%20%5Cfrac%7B-1%7D%20%7B3%7D%20x%5E2%20%5Ccos%203x%20%2B%20%5Cfrac%7B2%7D%20%7B9%7D%20x%20%5Csin%203x%20%2B%20%5Cfrac%7B2%7D%20%7B27%7D%20%5Ccos%203x%20%2B%20C." \* MERGEFORMATINET  Solving for an Unknown Integral There are some integrals that require us to evaluate two integrations by parts, followed by solving for the unknown integral. These kinds of integrals crop up often in electrical engineering and other disciplines. Example 6: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20e%5Ex%20%5Ccos%20x%20dx" \* MERGEFORMATINET . Solution: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20e%5Ex%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dv%20%3D%20%5Ccos%20x%20dx." \* MERGEFORMATINET Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20e%5Ex%20dx%2C%20v%20%3D%20%5Csin%20x%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20e%5Ex%20%5Ccos%20x%20dx%20%3D%20e%5Ex%20%5Csin%20x%20-%20%5Cint%20e%5Ex%20%5Csin%20x%20dx." \* MERGEFORMATINET  Notice that the second integral looks the same as our original integral in form, except that it has a  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%20x" \* MERGEFORMATINET instead of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ccos%20x." \* MERGEFORMATINET To evaluate it, we again apply integration by parts to the second term with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20e%5Ex%2C%20dv%20%3D%20%5Csin%20x%20dx%2C%20v%20%3D%20-%5Ccos%20x%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20e%5Ex%20dx." \* MERGEFORMATINET  Then  Notice that the unknown integral now appears on both sides of the equation. We can simply move the unknown integral on the right to the left side of the equation, thus adding it to our original integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=2%20%5Cint%20e%5Ex%20%5Ccos%20x%20dx%20%3D%20e%5Ex%20%5Csin%20x%20%2B%20e%5Ex%20%5Ccos%20x%20%2B%20C." \* MERGEFORMATINET  Dividing both sides by 2 we obtain  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20e%5Ex%20%5Ccos%20x%20dx%20%3D%20%5Cfrac%7B1%7D%20%7B2%7D%20e%5Ex%20%5Csin%20x%20%2B%20%5Cfrac%7B1%7D%20%7B2%7D%20e%5Ex%20%5Ccos%20x%20%2B%20%5Cfrac%7B1%7D%20%7B2%7D%20C." \* MERGEFORMATINET  Since the constant of integration is just a dummy constant, let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7BC%7D%20%7B2%7D%20%5Crightarrow%20C." \* MERGEFORMATINET  Finally, our solution is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20e%5Ex%20%5Ccos%20x%20dx%20%3D%20%5Cfrac%7B1%7D%20%7B2%7D%20e%5Ex%20%5Csin%20x%20%2B%20%5Cfrac%7B1%7D%20%7B2%7D%20e%5Ex%20%5Ccos%20x%20%2B%20C." \* MERGEFORMATINET  Multimedia Links To see this same "classic" example worked out with narration 17.0, see  HYPERLINK "http://www.youtube.com/watch?v=F-OsMq7QKEQ" Khan Academy Indefinite Integration Series Part 7 (9:39) INCLUDEPICTURE "http://i3.ytimg.com/vi/F-OsMq7QKEQ/hqdefault.jpg" \* MERGEFORMATINET . For additional video presentations on integration by parts 17.0, see  HYPERLINK "http://www.youtube.com/watch?v=815s-YB9X44" Math Video Tutorials by James Sousa, Integration by Parts, Basic (7:08) INCLUDEPICTURE "http://i1.ytimg.com/vi/815s-YB9X44/hqdefault.jpg" \* MERGEFORMATINET ;  HYPERLINK "http://www.youtube.com/watch?v=hfJ1zPizj_s" Math Video Tutorials by James Sousa, Integration by Parts (10:03) INCLUDEPICTURE "http://i1.ytimg.com/vi/hfJ1zPizj_s/hqdefault.jpg" \* MERGEFORMATINET ;  HYPERLINK "http://www.youtube.com/watch?v=_kJNhoTJ4D4" Math Video Tutorials by James Sousa, Integration by Parts, Additional Examples (7:47) INCLUDEPICTURE "http://i4.ytimg.com/vi/_kJNhoTJ4D4/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Evaluate the following integrals. (Remark: Integration by parts is not necessarily a requirement to solve the integrals. In some, you may need to use u-substitution along with integration by parts.)  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%203x%20e%5Ex%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5E2%20e%5E%7B-x%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cln%283x%20%2B%202%29dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%5E%7B-1%7D%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csec%5E3%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%202x%20%5Cln%283x%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B%28ln%20x%29%5E2%7D%20%7Bx%7D%20dx" \* MERGEFORMATINET  Use both the method of u-substitution and the method of integration by parts to integrate the integral below. Both methods will produce equivalent answers.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%20%5Csqrt%7B5x%20-%202%7D%20dx" \* MERGEFORMATINET  Use the method of tabular integration by parts to solve  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5E2%20e%5E%7B5x%7D%20dx." \* MERGEFORMATINET  Evaluate the definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E1%20x%5E2%20e%5Ex%20dx" \* MERGEFORMATINET . Evaluate the definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_1%5E3%20%5Cln%28x%20%2B%201%29%20dx" \* MERGEFORMATINET . Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=3xe%5E%7Bx%7D%20-3%20e%5E%7Bx%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20-e%5E%7B-x%7D%20%28x%5E2%20%2B%202x%20%2B%202%29%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B3x%20%2B%202%7D%20%7B3%7D%20%5B%5Cln%20%7C3x%20%2B%202%7C%20-%201%5D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Csin%5E%7B-1%7D%20x%20%2B%20%5Csqrt%7B1%20-%20x%5E2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D%20%28%5Csec%20x%29%20%28%5Ctan%20x%29%20%2B%20%5Cfrac%7B1%7D%20%7B2%7D%20%5Cln%20%7C%20%5Csec%20x%20%2B%20%5Ctan%20x%7C%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20x%5E2%20%5Cln%20%7C3x%7C%20-%20%5Cfrac%7B1%7D%20%7B2%7D%20x%5E2%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B3%7D%20%28%5Cln%20x%29%5E3%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2%7D%20%7B125%7D%20%285x%20-%202%29%5E%7B5/2%7D%20%2B%20%5Cfrac%7B4%7D%7B75%7D%285x%20-%202%29%5E%7B3/2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=e%5E%7B5x%7D%20%5Cleft%20%5B%20%5Cfrac%7Bx%5E2%7D%20%7B5%7D%20-%20%5Cfrac%7B2x%7D%20%7B25%7D%20%2B%20%5Cfrac%7B2%7D%20%7B125%7D%20%5Cright%5D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20e%20-%202" \* MERGEFORMATINET   EMBED Equation.DSMT4  Practice with Integration by Substitution & Integration by Parts Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Evaluate the definite integral. 13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  Answers: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  More Practice with Integration by Substitution & Integration by Parts Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Evaluate the definite integral. 13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  Answers: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  Integration by Parts with Trig HW Integrate! 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  Answers: 1.)  EMBED Equation.DSMT4  2.)  EMBED Equation.DSMT4  10.3 Integration by Partial Fractions Learning Objectives A student will be able to: Compute by hand the integrals of a wide variety of functions by using technique of Integration by Partial Fractions. Combine the technique of partial fractions with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution to solve various integrals. This is the third technique that we will study. This technique involves decomposing a rational function into a sum of two or more simple rational functions. For example, the rational function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7Bx%20%2B%204%7D%20%7Bx%5E2%20%2B%20x%20-%202%7D" \* MERGEFORMATINET  can be decomposed into  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7Bx%20%2B%204%7D%20%7Bx%5E2%20%2B%20x%20-%202%7D%20%3D%20%5Cfrac%7B2%7D%20%7Bx%20%2B%202%7D%20%2B%20%5Cfrac%7B3%7D%20%7Bx%20-%201%7D." \* MERGEFORMATINET  The two partial sums on the right are called partial factions. Suppose that we wish to integrate the rational function above. By decomposing it into two partial fractions, the integral becomes manageable:  To use this method, we must be able to factor the denominator of the original function and then decompose the rational function into two or more partial fractions. The examples below illustrate the method. Example 1: Find the partial fraction decomposition of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7B2x%20-%2019%7D%20%7Bx%5E2%20%2B%20x%20-%206%7D." \* MERGEFORMATINET  Solution: We begin by factoring the denominator as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%5E2%20%2B%20x%20-%206%20%3D%20%28x%20%2B%203%29%28x%20-%202%29." \* MERGEFORMATINET Then write the partial fraction decomposition as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7B2x%20-%2019%7D%20%7Bx%5E2%20%2B%20x%20-%206%7D%20%3D%20%5Cfrac%7BA%7D%20%7Bx%20%2B%203%7D%20%2B%20%5Cfrac%7BB%7D%20%7Bx%20-%202%7D." \* MERGEFORMATINET  Our goal at this point is to find the values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=B" \* MERGEFORMATINET . To solve this equation, multiply both sides of the equation by the factored denominator  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28x%20%2B%203%29%28x%20-%202%29." \* MERGEFORMATINET This process will produce the basic equation.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=2x%20-%2019%20%3D%20A%28x%20-%202%29%20%2B%20B%28x%20%2B%203%29." \* MERGEFORMATINET  This equation is true for all values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET The most convenient values are the ones that make a factor equal to zero, namely,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3." \* MERGEFORMATINET Substituting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=2%282%29-19%20%26%20%3D%20%20A%282-2%29%2B%20B%282%2B3%29%5C%5C%0A-15%20%26%20%3D%200%2B5B%5C%5C%0A-3%20%26%20%3D%20B" \* MERGEFORMATINET  Similarly, substituting for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-3" \* MERGEFORMATINET into the basic equation we get  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=2%28-3%29-19%20%26%20%3D%20%20A%28-3-2%29%2B%20B%28-3%2B3%29%5C%5C%0A-25%20%26%20%3D%20-5A%2B0%5C%5C%0A5%20%26%20%3D%20A" \* MERGEFORMATINET  We have solved the basic equation by finding the values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=B." \* MERGEFORMATINET Therefore, the partial fraction decomposition is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7B2x-19%7D%7Bx%5E2%2Bx-6%7D%3D%5Cfrac%20%7B5%7D%7Bx%2B3%7D-%5Cfrac%20%7B3%7D%7Bx-2%7D." \* MERGEFORMATINET  General Description of the Method To be able to write a rational function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29/g%28x%29" \* MERGEFORMATINET as a sum of partial fractions, must apply two conditions: The degree of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET must be less than the degree of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29." \* MERGEFORMATINET If so, the rational function is called proper. If it is not, divide  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29" \* MERGEFORMATINET (use long division) and work with the remainder term. The factors of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29" \* MERGEFORMATINET are known. If not, you need to find a way to find them. The guide below shows how you can write  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29/g%28x%29" \* MERGEFORMATINET as a sum of partial fractions if the factors of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29" \* MERGEFORMATINET are known. A Guide to Finding Partial Fractions Decomposition of a Rational Function 1. To find the partial fraction decomposition of a proper rational function,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29/g%28x%29%2C" \* MERGEFORMATINET factor the denominator  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28x%29" \* MERGEFORMATINET and write an equation that has the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7Bf%28x%29%7D%7Bg%28x%29%7D%20%3D%20%28%5Ctext%7Bsum%20of%20partial%20fractions%7D.%29" \* MERGEFORMATINET  2. For each distinct factor  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=ax%20%2B%20b%2C" \* MERGEFORMATINET the right side must include a term of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7BA%7D%7Bax%2Bb%7D." \* MERGEFORMATINET  3. For each repeated factor  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28ax%20%2B%20b%29%5En%2C" \* MERGEFORMATINET the right side must include n terms of the form  Example 2: Use the method of partial fractions to evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7Bx%2B1%7D%7B%28x%2B2%29%5E2%7D%20dx" \* MERGEFORMATINET . Solution: According to the guide above (item #3), we must assign the sum of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%202" \* MERGEFORMATINET  partial sums:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7Bx%2B1%7D%7B%28x%2B2%29%5E2%7D%3D%20%5Cfrac%20%7BA%7D%7B%28x%2B2%29%7D%20%2B%20%5Cfrac%20%7BB%7D%7B%28x%2B2%29%5E2%7D." \* MERGEFORMATINET  Multiply both sides by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28x%20%2B%202%29%5E2%3A" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=x%2B1%20%26%20%3D%20A%28x%2B2%29%20%2B%20B%20%5C%5C%0Ax%2B1%20%26%20%3D%20Ax%2B%282A%2BB%29." \* MERGEFORMATINET  Equating the coefficients of like terms from both sides,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=1%20%26%20%3D%20A%20%5C%5C%0A1%20%26%20%3D%202A%2BB." \* MERGEFORMATINET  Thus  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=A%20%26%20%3D%201.%20%5C%5C%0AB%20%26%20%3D%20-1." \* MERGEFORMATINET  Therefore the partial fraction decomposition is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7Bx%2B1%7D%7B%28x%2B2%29%5E2%7D%20%3D%20%20%5Cfrac%20%7B1%7D%7Bx%2B2%7D%20-%20%5Cfrac%20%7B1%7D%7B%28x%2B2%29%5E2%7D." \* MERGEFORMATINET  The integral will become  where we have used  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution for the second integral. Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7B3x%5E2%2B3x%2B1%7D%7Bx%5E3%2B2x%5E2%2Bx%7D%20dx" \* MERGEFORMATINET . Solution: We begin by factoring the denominator as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%28x%20%2B%201%29%5E2." \* MERGEFORMATINET Then the partial fraction decomposition is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7B3x%5E2%2B3x%2B1%7D%7Bx%5E3%2B2x%5E2%2Bx%7D%20%3D%20%5Cfrac%20%7BA%7D%7Bx%7D%20%2B%20%5Cfrac%20%7BB%7D%7Bx%2B1%7D%20%2B%20%5Cfrac%20%7BC%7D%7B%28x%2B1%29%5E2%7D." \* MERGEFORMATINET  Multiplying each side of the equation by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%28x%20%2B%201%29%5E2" \* MERGEFORMATINET we get the basic equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=3x%5E2%2B3x%2B1%20%3D%20A%28x%2B1%29%5E2%20%2B%20Bx%28x%2B1%29%20%2B%20Cx." \* MERGEFORMATINET  This equation is true for all values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET The most convenient values are the ones that make a factor equal to zero, namely,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET  Substituting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20-1%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=3%28-1%29%5E2%20%2B3%28-1%29%2B1%20%26%20%3D%20A%28-1%2B1%29%5E2%20%2B%20B%28-1%29%28-1%2B1%29%20%2B%20C%28-1%29%20%5C%5C%0A1%20%26%20%3D%20%200%20%2B%200%20-C%20%20%5C%5C%0A-1%20%26%20%3D%20%20C." \* MERGEFORMATINET  Substituting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET ,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=3%280%29%5E2%20%2B3%280%29%2B1%20%26%20%3D%20A%280%2B1%29%5E2%20%2B%20B%280%29%280%2B1%29%20%2B%20C%280%29%20%5C%5C%0A1%20%26%20%3D%20A%2B0%2B0%20%5C%5C%0A1%20%26%20%3D%20A." \* MERGEFORMATINET  To find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=B" \* MERGEFORMATINET we can simply substitute any value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET along with the values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C" \* MERGEFORMATINET obtained. Choose  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET :  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=3%281%29%5E2%20%2B3%281%29%2B1%20%26%20%3D%20%20A%281%2B1%29%5E2%20%2B%20B%281%29%281%2B1%29%20%2B%20C%281%29%20%5C%5C%0A7%20%20%26%20%3D%20%204%20%2B%202B%20-1%20%20%5C%5C%0A2%20%20%26%20%3D%20%20B." \* MERGEFORMATINET  Now we have solved for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%2C%20B%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C." \* MERGEFORMATINET We use the partial fraction decomposition to integrate.  Example 4: This problem is an example of an improper rational function. Evaluate the definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_%7B1%7D%5E%7B2%7D%20%5Cfrac%20%7Bx%5E3-4x%5E2-3x%2B3%7D%7Bx%5E2-3x%7D%20dx." \* MERGEFORMATINET  Solution: This rational function is improper because its numerator has a degree that is higher than its denominator. The first step is to divide the denominator into the numerator by long division and obtain  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7Bx%5E3-4x%5E2-3x%2B3%7D%7Bx%5E2-3x%7D%20%3D%20%7B%28x-1%29%7D%20%2B%20%5Cfrac%20%7B-6x%2B3%7D%7Bx%5E2-3x%7D." \* MERGEFORMATINET  Now apply partial function decomposition only on the remainder,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%20%7B-6x%2B3%7D%7Bx%5E2-3x%7D%20%3D%20%5Cfrac%20%7B-6x%2B3%7D%7Bx%28x-3%29%7D%20%3D%20%5Cfrac%20%7BA%7D%7Bx%7D%20%2B%20%5Cfrac%20%7BB%7D%7Bx-3%7D." \* MERGEFORMATINET  As we did in the previous examples, multiply both sides by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%28x-3%29" \* MERGEFORMATINET and then set  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203" \* MERGEFORMATINET to obtain the basic equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=-6x%2B3%3D%20A%28x-3%29%2B%20Bx" \* MERGEFORMATINET  For  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=3%20%26%20%3D%20%20-3A%2B0%5C%5C%0A-1%20%26%20%3D%20%20A." \* MERGEFORMATINET  For  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=-18%2B3%20%26%20%3D%200%2B3B%20%5C%5C%0A-15%20%26%20%3D%203B%20%5C%5C%0A-5%20%26%20%3D%20B." \* MERGEFORMATINET  Thus our integral becomes  Integrating and substituting the limits,  Multimedia Links For a complete partial fractions problem (19.0), see  HYPERLINK "http://www.youtube.com/watch?v=9AXr7dIr0Ng" Integration by Partial Fractions, Just Math Tutoring (6:02) INCLUDEPICTURE "http://i2.ytimg.com/vi/9AXr7dIr0Ng/hqdefault.jpg" \* MERGEFORMATINET  and for integration using partial fractions and a rationalizing substitution (19.0), see  HYPERLINK "http://www.youtube.com/watch?v=D8OncQ9-dHY" Integration using Partial Fractions and a rationalizing substitution, Just Math Tutoring (6:06) INCLUDEPICTURE "http://i1.ytimg.com/vi/D8OncQ9-dHY/hqdefault.jpg" \* MERGEFORMATINET . For an extensive presentation on integrating with partial fractions including the completing the square technique (19.0) see  HYPERLINK "http://www.youtube.com/watch?v=aeXp1zC6Hls" Integration with partial fractions using various techniques, MIT Courseware (51:24) INCLUDEPICTURE "http://i2.ytimg.com/vi/aeXp1zC6Hls/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Evaluate the following integrals.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7B1%7D%7Bx%5E2%20-%201%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7Bx%7D%7Bx%5E2%20-%202x%20-%203%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7B1%7D%7Bx%5E3%20%2B%20x%5E2-2x%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7Bx%5E3%7D%7Bx%5E2%20%2B%204%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%20%7B%5Cphi%7D%7B1%20%2B%20%7B%5Cphi%7D%7D%20d%7B%5Cphi%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B5%7D%20%5Cfrac%20%7Bx-1%7D%7Bx%5E2%28x%20%2B%201%29%7D%20dx" \* MERGEFORMATINET  Evaluate the integral by making the proper u-substitution to convert to a rational function:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7Bcos%7B%5Ctheta%7D%7D%7B%5Csin%5E2%20%7B%5Ctheta%7D%2B%204%5Csin%20%7B%5Ctheta%7D%20-5%7D%20d%7B%5Ctheta%7D." \* MERGEFORMATINET  Evaluate the integral by making the proper u-substitution to convert to a rational function:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7B3e%5E%7B%5Ctheta%7D%7D%7Be%5E%7B2%20%5Ctheta%7D-1%7D%20d%7B%5Ctheta%7D." \* MERGEFORMATINET  Find the area under the curve  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%201/%282%20%2B%20e%5Ex%29%2C" \* MERGEFORMATINET  over the interval [-ln 3, ln 4]. (Hint: make a u-substitution to convert the integrand into a rational function.) Show that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7B1%7D%7Ba%5E2-x%5E2%7Ddx%20%3D%20%5Cfrac%20%7B1%7D%7B2a%7D%20%7B%20%5Cln%7D%7B%5Cleft%7C%5Cfrac%7Ba%20%2B%20x%7D%7Ba%20-%20x%7D%5Cright%7C%7D%20%2B%20C." \* MERGEFORMATINET  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B2%7D%20%7B%20%5Cln%7D%5Cbegin%7Bvmatrix%7D%7B%5Cfrac%7Bx-1%7D%7Bx%2B1%7D%7D%20%5Cend%7Bvmatrix%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B4%7D%20%7B%20%5Cln%7D%5Cbegin%7Bvmatrix%7D%7Bx%2B1%7D%20%5Cend%7Bvmatrix%7D%20%2B%20%5Cfrac%20%7B3%7D%7B4%7D%20%7B%20%5Cln%7D%5Cbegin%7Bvmatrix%7D%7Bx-3%7D%20%5Cend%7Bvmatrix%7D%20%2B%20C" \* MERGEFORMATINET   EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B2%7D%20x%5E2%20-%202%20%7B%20%5Cln%7D%28x%5E2%2B4%29%20%2BC" \* MERGEFORMATINET  1 ln 2  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B-4%7D%7B5%7D%20%2B%202%20%7B%20%5Cln%7D%5Cfrac%20%7B5%7D%7B3%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B6%7D%20%7B%20%5Cln%7D%5Cleft%20%28%20%5Cfrac%7B%5Csin%20%7B%5Ctheta%7D-1%7D%7B5%2B%5Csin%20%7B%5Ctheta%7D%7D%20%5Cright%20%29%20%2B%20C" \* MERGEFORMATINET   EMBED Equation.DSMT4   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B2%7D%20%7B%20%5Cln%7D%5Cfrac%20%7B14%7D%7B3%7D" \* MERGEFORMATINET  Hint: Decompose the integrand into partial fractions. 10.4 Trigonometric Integrals Learning Objectives A student will be able to: Compute by hand the integrals of a wide variety of functions by using the Trigonometric Integrals. Combine this technique with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution. Integrating Powers of Sines and Cosines In this section we will study methods of integrating functions of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%7B%5Csin%5Em%20%7Bx%7D%7D%20%20%7B%5Ccos%5En%20%7Bx%7D%7D%20dx%2C" \* MERGEFORMATINET  where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET are nonnegative integers. The method that we will describe uses the famous trigonometric identities  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7B%5Csin%5E2%7Bx%7D%7D%20%3D%20%5Cfrac%20%7B1%7D%7B2%7D%20%281-%5Ccos%20%7B2x%7D%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7B%5Ccos%5E2%7Bx%7D%7D%20%3D%20%5Cfrac%20%7B1%7D%7B2%7D%20%281%2B%5Ccos%20%7B2x%7D%29." \* MERGEFORMATINET  Example 1: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%7B%5Csin%5E2%20%7Bx%7D%7D%20dx" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%7B%5Ccos%5E2%20%7Bx%7D%7D%20dx." \* MERGEFORMATINET  Solution: Using the identities above, the first integral can be written as  Similarly, the second integral can be written as  Example 2: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ccos%5E4%20x%20dx." \* MERGEFORMATINET  Solution:  Integrating term by term,  Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%5E3%20x%20dx." \* MERGEFORMATINET  Solution:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Csin%5E3%20x%20dx%20%3D%20%5Cint%20%5Csin%5E2%20x%20%5Csin%20x%20dx" \* MERGEFORMATINET  Recall that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Csin%5E2%20x%20%2B%20%5Ccos%5E2%20x%20%3D%201%2C" \* MERGEFORMATINET  so by substitution,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%26%20%3D%20%5Cint%20%281%20-%20%5Ccos%5E2%20x%29%20%5Csin%20x%20dx%5C%5C%0A%26%20%3D%20%5Cint%20%5Csin%20x%20dx%20-%20%5Cint%20%5Ccos%5E2%20x%20%5Csin%20x%20dx." \* MERGEFORMATINET  The first integral should be straightforward. The second can be done by the method of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution by letting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ccos%20x%2C" \* MERGEFORMATINET so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20-%5Csin%20x%20dx." \* MERGEFORMATINET The integral becomes  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET are both positive integers, then an integral of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Csin%5Em%20x%20%5Ccos%5En%20x%20dx" \* MERGEFORMATINET  can be evaluated by one of the procedures shown in the table below, depending on whether  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET are odd or even.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7B%5Cint%20%5Csin%5Em%20x%20%5Ccos%5En%20x%20dx%7D" \* MERGEFORMATINET Procedure Identities  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET odd Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Csin%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ccos%5E2%20x%20%3D%201%20-%20%5Csin%5E2%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET odd Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ccos%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%5E2%20x%20%3D%201%20-%20%5Ccos%5E2%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET even Use identities to reduce powers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%5E2%20x%20%3D%20%281/2%29%20%281%20-%20%5Ccos%202x%29" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ccos%5E2%20x%20%3D%20%281/2%29%20%281%20%2B%20%5Ccos%202x%29" \* MERGEFORMATINET Example 4: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%5E3%20x%20%5Ccos%5E4%20x%20dx." \* MERGEFORMATINET  Solution: Here,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET is odd. So according to the second procedure in the table above, let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ccos%20x%2C" \* MERGEFORMATINET so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20-%5Csin%20x." \* MERGEFORMATINET Substituting,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Csin%5E3%20x%20%5Ccos%5E4%20x%20dx%20%26%20%3D%20%5Cint%20u%5E4%20%5Csin%5E3%20x%20%5Cfrac%7B-1%7D%20%7B%5Csin%20x%7D%20du%5C%5C%0A%26%20%3D%20-%5Cint%20u%5E4%20%5Csin%5E2%20x%20du." \* MERGEFORMATINET  Referring to the table again, we can now substitute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%5E2%20x%20%3D%201%20-%20%5Ccos%5E2%20x" \* MERGEFORMATINET in the integral:  Example 5: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%5E4%20x%20%5Ccos%5E4%20x%20dx" \* MERGEFORMATINET . Solution: Here,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m%20%3D%20n%20%3D%204." \* MERGEFORMATINET We follow the third procedure in the table above:  At this stage, it is best to use  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution to integrate. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%202x%2C" \* MERGEFORMATINET so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%202dx." \* MERGEFORMATINET   Integrating Powers of Secants and Tangents In this section we will study methods of integrating functions of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%5Em%20x%20%5Csec%5En%20x%20dx%2C" \* MERGEFORMATINET  where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET are nonnegative integers. However, we will begin with the integrals  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%20x%20dx" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Csec%20x%20dx." \* MERGEFORMATINET  The first integral can be evaluated by writing  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%20x%20dx%20%3D%20%5Cint%20%5Cfrac%7B%5Csin%20x%7D%20%7B%5Ccos%20x%7D%20dx." \* MERGEFORMATINET  Using  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution, let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ccos%20x%2C" \* MERGEFORMATINET so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20-%5Csin%20x%20dx." \* MERGEFORMATINET The integral becomes  The second integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csec%20x%20dx" \* MERGEFORMATINET , however, is not straightforwardit requires a trick. Let  Use  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Csec%20x%20%2B%20%5Ctan%20x%2C" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%3D%20%28%5Csec%5E2%20x%20%2B%20%5Csec%20x%20%5Ctan%20x%29%20dx%2C" \* MERGEFORMATINET the integral becomes,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Csec%20x%20dx%20%26%3D%20%5Cint%20%5Cfrac%7Bdu%7D%20%7Bu%7D%5C%5C%0A%26%3D%20%5Cln%20%7Cu%7C%20%2B%20C%5C%5C%0A%26%3D%20%5Cln%20%7C%5Csec%20x%20%2B%20%5Ctan%20x%7C%20%2B%20C." \* MERGEFORMATINET  There are two reduction formulas that help evaluate higher powers of tangent and secant:  Example 6: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csec%5E3%20x%20dx" \* MERGEFORMATINET . Solution: We use the formula above by substituting for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%203." \* MERGEFORMATINET   Example 7: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ctan%5E5%20x%20dx" \* MERGEFORMATINET . Solution: We use the formula above by substituting for  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m%20%3D%205." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%5E5%20x%20dx%20%3D%20%5Cfrac%7B%5Ctan%5E4%20x%7D%20%7B4%7D%20-%20%5Cint%20%5Ctan%5E3%20x%20dx." \* MERGEFORMATINET  We need to use the formula again to solve the integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ctan%5E3%20x%20dx" \* MERGEFORMATINET :  If  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET are both positive integers, then an integral of the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%5Em%20x%20%5Csec%5En%20x%20dx" \* MERGEFORMATINET  can be evaluated by one of the procedures shown in the table below, depending on whether  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET are odd or even.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7B%5Cint%20%5Ctan%5Em%20x%20%5Csec%5En%20x%20dx%7D" \* MERGEFORMATINET Procedure Identities  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET even Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%3D%20%5Ctan%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csec%5E2%20x%20%3D%20%5Ctan%5E2%20x%20%2B%201" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET odd Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Csec%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctan%5E2%20x%20%3D%20%5Csec%5E2%20x%20-%201" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m" \* MERGEFORMATINET even  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n" \* MERGEFORMATINET odd Reduce powers of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csec%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctan%5E2x%20%3D%20%5Csec%5E2x%20-%201" \* MERGEFORMATINET Example 8: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ctan%5E2%20x%20%5Csec%5E4%20x%20dx" \* MERGEFORMATINET . Solution: Here  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=n%20%3D%204" \* MERGEFORMATINET is even, and so we will follow the first procedure in the table above. Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Ctan%20x%2C" \* MERGEFORMATINET so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20%5Csec%5E2%20x%20dx." \* MERGEFORMATINET Before we substitute, split off a factor of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csec%5E2%20x." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%5E2%20x%20%5Csec%5E4%20x%20dx%20%3D%20%5Cint%20%5Ctan%5E2%20x%20%5Csec%5E2%20x%20%5Csec%5E2%20x%20dx." \* MERGEFORMATINET  Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csec%5E2%20x%20%3D%20%5Ctan%5E2%20x%20%2B%201%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%3D%20%5Cint%20%5Ctan%5E2%20x%20%28%5Ctan%5E2%20x%20%2B%201%29%20%5Csec%5E2%20x%20dx." \* MERGEFORMATINET  Now we make the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution:  Example 9: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ctan%5E3%20x%20%5Csec%5E3%20x%20dx" \* MERGEFORMATINET . Solution: Here  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m%3D%203" \* MERGEFORMATINET is odd. We follow the third procedure in the table. Make the substitution,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Csec%20x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=du%20%3D%20%5Csec%20x%20%5Ctan%20x%20dx." \* MERGEFORMATINET Our integral becomes  Multimedia Links For video presentations on computing the integrals of trigonometric functions (20.0), see  HYPERLINK "http://www.youtube.com/watch?v=lTqnlihOC4o" Trigonometric Integrals, Part 1 (5:57) ; see  HYPERLINK "http://www.youtube.com/watch?v=zyg9k1je7Fg" Trigonometric Integrals, Part 2 (6:01) INCLUDEPICTURE "http://i3.ytimg.com/vi/zyg9k1je7Fg/hqdefault.jpg" \* MERGEFORMATINET ; see  HYPERLINK "http://www.youtube.com/watch?v=BhJ4soojyAQ" Trigonometric Integrals, Part 3 (5:54) INCLUDEPICTURE "http://i3.ytimg.com/vi/BhJ4soojyAQ/hqdefault.jpg" \* MERGEFORMATINET ; see  HYPERLINK "http://www.youtube.com/watch?v=8CHHY-2Ctug" Trigonometric Integrals, Part 4 (8:57) INCLUDEPICTURE "http://i1.ytimg.com/vi/8CHHY-2Ctug/hqdefault.jpg" \* MERGEFORMATINET ; see  HYPERLINK "http://www.youtube.com/watch?v=QdNScjd5bno" Trigonometric Integrals, Part 5 (5:58) INCLUDEPICTURE "http://i2.ytimg.com/vi/QdNScjd5bno/hqdefault.jpg" \* MERGEFORMATINET ; see  HYPERLINK "http://www.youtube.com/watch?v=m3zG7c52QR4" Trigonometric Integrals, Part 6 (8:42) INCLUDEPICTURE "http://i2.ytimg.com/vi/m3zG7c52QR4/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Evaluate the integrals.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ccos%5E4%20x%20%5Csin%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%5E2%205%5Cphi%20d%5Cphi" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%5E2%202z%20%5Ccos%5E3%202z%20dz%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csin%20x%20%5Ccos%28x/2%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csec%5E4%20x%20%5Ctan%5E3%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ctan%5E4%20x%20%5Csec%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csqrt%7B%5Ctan%20x%7D%20%5Csec%5E4%20x%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E%7B%5Cpi/2%7D%20%5Ctan%5E5%20%5Cfrac%7Bx%7D%20%7B2%7D%20dx" \* MERGEFORMATINET  Graph and then find the volume of the solid that results when the region enclosed by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20%5Csin%20x%2C%20y%20%3D%20%5Ccos%20x%2C%20x%20%3D%200%2C" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Cpi/4" \* MERGEFORMATINET  is revolved around the x-axis. Prove that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ccsc%20x%20dx%20%3D%20-%5Cln%20%7C%5Ccsc%20x%20%2B%20%5Ccot%20x%7C%20%2B%20C" \* MERGEFORMATINET  Show that it can also be written in the following two forms:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Ccsc%20x%20dx%20%3D%20%5Cln%20%7C%5Ctan%20%5Cfrac%7B1%7D%20%7B2%7D%20x%7C%20%2B%20C%20%3D%20%5Cln%20%7C%5Ccsc%20x%20-%20%5Ccot%20x%7C%20%2B%20C." \* MERGEFORMATINET  Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B5%7D%20%5Ccos%5E5%20x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D%5Cphi%20-%20%5Cfrac%7B1%7D%20%7B20%7D%20%5Csin%2010%5Cphi%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B6%7D%20%5Csin%5E3%202z%20-%20%5Cfrac%7B1%7D%20%7B10%7D%20%5Csin%5E5%202z%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-1%7D%20%7B3%7D%20%5Ccos%28%5Cfrac%7B3x%7D%20%7B2%7D%20%29%20-%20%5Ccos%28%5Cfrac%7Bx%7D%20%7B2%7D%29%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B7%7D%20%5Csec%5E7%20x%20-%20%5Cfrac%7B1%7D%20%7B5%7D%20%5Csec%5E5%20x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B4%7D%20%5Csec%5E3%20x%20%5Ctan%20x%20-%20%5Cfrac%7B5%7D%20%7B8%7D%20%5Csec%20x%20%5Ctan%20x%20%2B%20%5Cfrac%7B3%7D%20%7B8%7D%20%5Cln%20%7C%5Csec%20x%20%2B%20%5Ctan%20x%7C%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2%7D%20%7B3%7D%20%5Ctan%5E%7B3/2%7D%20x%20%2B%20%5Cfrac%7B2%7D%20%7B7%7D%20%5Ctan%5E%7B7/2%7D%20x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-1%7D%20%7B2%7D%20%2B%20%5Cln%202" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20V%20%3D%20%5Cpi/2" \* MERGEFORMATINET  . 10.5 Trigonometric Substitutions Learning Objectives A student will be able to: Compute by hand the integrals of a wide variety of functions by using technique of Trigonometric Substitution. Combine this technique with other integration techniques to integrate. When we are faced with integrals that involve radicals of the forms  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7Ba%5E2%20-%20x%5E2%7D%2C%20%5Csqrt%7Bx%5E2%20-%20a%5E2%7D%2C" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7Bx%5E2%20%2B%20a%5E2%7D%2C" \* MERGEFORMATINET we may make substitutions that involve trigonometric functions to eliminate the radical. For example, to eliminate the radical in the expression  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Csqrt%7Ba%5E2%20-%20x%5E2%7D" \* MERGEFORMATINET  we can make the substitution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=x%20%3D%20a%20%5Csin%5Ctheta%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=-%5Cpi/2%20%5Cle%20%5Ctheta%20%5Cle%20%5Cpi/2%2C" \* MERGEFORMATINET  (Note:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctheta" \* MERGEFORMATINET  must be limited to the range of the inverse sine function.) which yields,  The reason for the restriction  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-%5Cpi/2%20%5Cle%20%5Ctheta%20%5Cle%20%5Cpi/2" \* MERGEFORMATINET  is to guarantee that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%20%5Ctheta" \* MERGEFORMATINET is a one-to-one function on this interval and thus has an inverse. The table below lists the proper trigonometric substitutions that will enable us to integrate functions with radical expressions in the forms above. Expression in Integrand Substitution Identity Needed  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7Ba%5E2%20-%20x%5E2%7D" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a%20%5Csin%20%5Ctheta" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1%20-%20%5Csin%5E2%20%5Ctheta%20%3D%20%5Ccos%5E2%20%5Ctheta" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7Ba%5E2%20%2B%20x%5E2%7D" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a%20%5Ctan%20%5Ctheta" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1%20%2B%20%5Ctan%5E2%20%5Ctheta%20%3D%20%5Csec%5E2%20x" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7Bx%5E2%20-%20a%5E2%7D" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20a%20%5Csec%20%5Ctheta" \* MERGEFORMATINET  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=sec%5E2%20%5Ctheta%20-%201%20%3D%20%5Ctan%5E2%20%5Ctheta" \* MERGEFORMATINET In the second column are listed the most common substitutions. They come from the reference right triangles, as shown in the figure below. We want any of the substitutions we use in the integration to be reversible so we can change back to the original variable afterward. The right triangles in the figure below will help us reverse our substitutions.  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123847" \* MERGEFORMATINET  Description: 3 triangles. Example 1: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7Bdx%7D%20%7Bx%5E2%20%5Csqrt%7B4%20-%20x%5E2%7D%7D." \* MERGEFORMATINET  Solution: Our goal first is to eliminate the radical. To do so, look up the table above and make the substitution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=x%20%3D%202%20%5Csin%20%5Ctheta%2C%20-%5Cpi/2%20%5Cle%20%5Ctheta%20%5Cle%20%5Cpi/2%2C" \* MERGEFORMATINET  so that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cfrac%7Bdx%7D%20%7Bd%5Ctheta%7D%20%3D%202%20%5Ccos%20%5Ctheta" \* MERGEFORMATINET  Our integral becomes  Up to this stage, we are done integrating. To complete the solution however, we need to express  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ccot%5Ctheta" \* MERGEFORMATINET in terms of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x." \* MERGEFORMATINET Looking at the figure of triangles above, we can see that the second triangle represents our case, with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a%20%3D%202." \* MERGEFORMATINET So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202%20%5Csin%20%5Ctheta" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2%20%5Ccos%20%5Ctheta%20%3D%20%5Csqrt%7B4%20-%20x%5E2%7D" \* MERGEFORMATINET , thus  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Ccot%5Ctheta%20%3D%20%5Cfrac%7B%5Csqrt%7B4%20-%20x%5E2%7D%7D%20%7Bx%7D%2C" \* MERGEFORMATINET  since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Ccot%5Ctheta%20%3D%20%5Cfrac%7B%5Ccos%20%5Ctheta%7D%20%7B%5Csin%20%5Ctheta%7D." \* MERGEFORMATINET  so that  Example 2: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B%5Csqrt%7Bx%5E2%20-%203%7D%7D%20%7Bx%7D%20dx" \* MERGEFORMATINET . Solution: Again, we want to first to eliminate the radical. Consult the table above and substitute  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Csqrt%7B3%7D%20%5Csec%20%5Ctheta" \* MERGEFORMATINET . Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%20%3D%20%5Csqrt%7B3%7D%20%5Csec%20%5Ctheta%20%20%5Ctan%20%5Ctheta%20d%5Ctheta" \* MERGEFORMATINET . Substituting back into the integral,  Using the integral identity from the section on Trigonometric Integrals,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Ctan%5Em%20x%20dx%20%3D%20%5Cfrac%7B%5Ctan%5E%7Bm%20-%201%7D%20x%7D%20%7Bm%20-%201%7D%20-%20%5Cint%20%5Ctan%5E%7Bm%20-%202%7D%20x%20dx." \* MERGEFORMATINET  and letting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=m%20%3D%202" \* MERGEFORMATINET we obtain  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7B%5Csqrt%7Bx%5E2%20-%203%7D%7D%20%7Bx%7D%20dx%20%3D%20%5Csqrt%7B3%7D%20%5Ctan%20%5Ctheta%20-%20%5Ctheta%20%2B%20C." \* MERGEFORMATINET  Looking at the triangles above, the third triangle represents our case, with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a%20%3D%20%5Csqrt%7B3%7D" \* MERGEFORMATINET . So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Csqrt%7B3%7D%20%5Csec%20%5Ctheta" \* MERGEFORMATINET and thus  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ccos%20x%20%3D%20%5Csqrt%7B3%7D/x" \* MERGEFORMATINET , which gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctan%20%5Ctheta%20%3D%20%5Csqrt%7Bx%5E2%20-%203%7D/%5Csqrt%7B3%7D" \* MERGEFORMATINET . Substituting,  Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7Bdx%7D%7Bx%5E%7B2%7D%20%5Csqrt%7Bx%5E%7B2%7D%2B1%7D%7D" \* MERGEFORMATINET . Solution: From the table above, let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Ctan%20%5Ctheta" \* MERGEFORMATINET then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=dx%20%3D%20%5Csec%5E2%5Ctheta%20d%5Ctheta." \* MERGEFORMATINET Substituting into the integral,  But since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ctan%5E2%5Ctheta%20%2B%201%20%3D%20%5Csec%5E2%5Ctheta%2C" \* MERGEFORMATINET   Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bd%7D%20%7Bd%5Ctheta%7D%20%28%5Ccsc%20%5Ctheta%29%20%3D%20-%5Ccot%5Ctheta%20%5Ccsc%20%5Ctheta%2C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7Bdx%7D%20%7Bx%5E2%20%5Csqrt%7Bx%5E2%20%2B%201%7D%7D%20%26%20%3D%20%5Cint%20%5Ccot%5Ctheta%20%5Ccsc%20%5Ctheta%20d%5Ctheta%5C%5C%0A%26%20%3D%20-%5Ccsc%20%5Ctheta%20%2B%20C." \* MERGEFORMATINET  Looking at the triangles above, the first triangle represents our case, with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=a%20%3D%201." \* MERGEFORMATINET So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20%5Ctan%20%5Ctheta%20" \* MERGEFORMATINET and thus  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Csin%20x%20%3D%20%5Cfrac%7Bx%7D%20%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%2C" \* MERGEFORMATINET which gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ccsc%20%5Ctheta%20%3D%20%5Cfrac%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%20%7Bx%7D." \* MERGEFORMATINET Substituting,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%7Bdx%7D%20%7Bx%5E2%20%5Csqrt%7Bx%5E2%20%2B%201%7D%7D%20%26%20%3D%20-%5Ccsc%20%5Ctheta%20%2B%20C%5C%5C%0A%26%20%3D%20-%20%5Cfrac%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%20%7Bx%7D%20%2B%20C." \* MERGEFORMATINET  Multimedia Links For video presentations on Trigonometric Substitutions (17.0), see  HYPERLINK "http://www.youtube.com/watch?v=yW6Odu0YHL0" Trigonometric Substitutions, Just Math Tutoring (9:30) INCLUDEPICTURE "http://i2.ytimg.com/vi/yW6Odu0YHL0/hqdefault.jpg" \* MERGEFORMATINET  and  HYPERLINK "http://www.youtube.com/watch?v=hxt5hd0CFXc" Trigonometric Substitutions, Part 2, Just Math Tutoring (4:20) INCLUDEPICTURE "http://i1.ytimg.com/vi/hxt5hd0CFXc/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Evaluate the integrals.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Csqrt%7B4%20-%20x%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B1%7D%20%7B%5Csqrt%7B9%20%2B%20x%5E2%20%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7Bx%5E3%7D%20%7B%5Csqrt%7B1%20-%20x%5E2%7D%7D%20dx%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B1%7D%20%7B%5Csqrt%7B1%20-%209x%5E2%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%5E3%20%5Csqrt%7B4%20-%20x%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B1%7D%20%7Bx%5E2%20%5Csqrt%7Bx%5E2%20-%2036%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%7B1%7D%20%7B%28x%5E2%20%2B%2025%29%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_0%5E4%20x%5E3%20%5Csqrt%7B16%20-%20x%5E2%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B-%5Cpi%7D%5E%7B0%7D%20e%5Ex%20%5Csqrt%7B1%20-%20e%5E%7B2x%7D%7D%20dx" \* MERGEFORMATINET  (Hint: First use u-substitution, letting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20e%5Ex" \* MERGEFORMATINET ) Graph and then find the area of the surface generated by the curve  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20x%5E2" \* MERGEFORMATINET  from x = 0 to x = 1 and revolved about the x-axis. Review Answers  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2%20%5Csin%5E%7B-1%7D%20%5Cleft%20%28%20%5Cfrac%7Bx%7D%20%7B2%7D%5Cright%20%29%20%2B%20%5Cfrac%7B1%7D%20%7B2%7Dx%20%5Csqrt%7B4%20-%20x%5E2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%20%28%5Cfrac%7B1%7D%7B3%7D%7C%5Csqrt%7B9%20%2B%20x%5E2%7D%20%2B%20x%7C%29%20%2B%20C%20" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B-x%5E2%20%5Csqrt%7B1%20-%20x%5E2%7D%7D%20%7B3%7D%20-%20%5Cfrac%7B2%7D%20%7B3%7D%20%5Csqrt%7B1%20-%20x%5E2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B3%7D%20%5Csin%5E%7B-1%7D%203x%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-%20%5Cfrac%7B4%7D%20%7B3%7D%20%284%20-%20x%5E2%29%5E%7B3/2%7D%20%2B%20%5Cfrac%7B1%7D%20%7B5%7D%20%284%20-%20x%5E2%29%5E%7B5/2%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B36%7D%20%5Cfrac%7B%5Csqrt%7Bx%5E2%20-%2036%7D%7D%20%7Bx%7D%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B50%7D%20%5Cfrac%7Bx%7D%20%7Bx%5E2%20%2B%2025%7D%20%2B%20%5Cfrac%7B1%7D%20%7B250%7D%20%5Ctan%5E%7B-1%7D%20%28x/5%29%20%2B%20C" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2048%7D%20%7B15%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D%20%5Cleft%20%5B%5Cfrac%7B%5Cpi%7D%20%7B2%7D-%5Csin%5E%7B-1%7D%20%28e%5E%7B-%5Cpi%7D%29%20-%20e%5E%7B-%5Cpi%7D%20%5Csqrt%7B1%20-%20e%5E%7B-2%5Cpi%7D%7D%20%20%5Cright%5D" \* MERGEFORMATINET  Surface area is  EMBED Equation.DSMT4  10.6 Improper Integrals Learning Objectives A student will be able to: Compute by hand the integrals of a wide variety of functions by using the technique of Improper Integration. Combine this technique with other integration techniques to integrate. Distinguish between proper and improper integrals. The concept of improper integrals is an extension to the concept of definite integrals. The reason for the term improper is because those integrals either include integration over infinite limits or the integrand may become infinite within the limits of integration. We will take each case separately. Recall that in the definition of definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_a%5Eb%20f%28x%29%20dx" \* MERGEFORMATINET we assume that the interval of integration  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D" \* MERGEFORMATINET is finite and the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous on this interval. Integration Over Infinite Limits If the integrand  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is continuous over the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20%5Cinfty%29%2C" \* MERGEFORMATINET then the improper integral in this case is defined as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_a%5E%7B%5Cinfty%7D%20f%28x%29%20dx%20%3D%20%5Clim_%7Bl%20%5Cto%20%5Cinfty%7D%20%5Cint_a%5El%20f%28x%29%20dx." \* MERGEFORMATINET  If the integration of the improper integral exists, then we say that it converges. But if the limit of integration fails to exist, then the improper integral is said to diverge. The integral above has an important geometric interpretation that you need to keep in mind. Recall that, geometrically, the definite integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_a%5Eb%20f%28x%29%20dx" \* MERGEFORMATINET represents the area under the curve. Similarly, the integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_a%5El%20f%28x%29%20dx" \* MERGEFORMATINET is a definite integral that represents the area under the curve  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET over the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20l%5D%2C" \* MERGEFORMATINET as the figure below shows. However, as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=l" \* MERGEFORMATINET approaches  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cinfty" \* MERGEFORMATINET , this area will expand to the area under the curve of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET and over the entire interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20%5Cinfty%29." \* MERGEFORMATINET Therefore, the improper integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_a%5E%7B%5Cinfty%7D%20f%28x%29%20dx" \* MERGEFORMATINET can be thought of as the area under the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f%28x%29" \* MERGEFORMATINET over the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20%5Cinfty%29." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123845" \* MERGEFORMATINET  Example 1: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_1%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%20%7Bx%7D" \* MERGEFORMATINET . Solution: We notice immediately that the integral is an improper integral because the upper limit of integration approaches infinity. First, replace the infinite upper limit by the finite limit  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=l" \* MERGEFORMATINET and take the limit of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=l" \* MERGEFORMATINET  to approach infinity:  Thus the integral diverges. Example 2: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_2%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%20%7Bx%5E2%7D" \* MERGEFORMATINET . Solution:  Thus the integration converges to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%7D%20%7B2%7D." \* MERGEFORMATINET  Example 3: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B%2B%20%5Cinfty%7D%5E%7B-%20%5Cinfty%7D%20%5Cfrac%7Bdx%7D%20%7B1%20%2B%20x%5E2%7D" \* MERGEFORMATINET . Solution: What we need to do first is to split the integral into two intervals  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28-%5Cinfty%2C%200%5D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%20%2B%5Cinfty%29." \* MERGEFORMATINET So the integral becomes  Next, evaluate each improper integral separately. Evaluating the first integral on the right,  Evaluating the second integral on the right,  Adding the two results,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_%7B-%5Cinfty%7D%5E%7B%2B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%20%7B1%20%2B%20x%5E2%7D%20%26%20%3D%20%5Cfrac%7B%5Cpi%7D%20%7B2%7D%20%2B%20%5Cfrac%7B%5Cpi%7D%20%7B2%7D%20%3D%20%5Cpi." \* MERGEFORMATINET  Remark: In the previous example, we split the integral at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET However, we could have split the integral at any value of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%20c" \* MERGEFORMATINET  without affecting the convergence or divergence of the integral. The choice is completely arbitrary. This is a famous theorem that we will not prove here. That is,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_%7B-%5Cinfty%7D%5E%7B%2B%5Cinfty%7D%20f%28x%29%20dx%20%3D%20%5Cint_%7B-%5Cinfty%7D%5Ec%20f%28x%29%20dx%20%2B%20%5Cint_c%5E%7B%2B%5Cinfty%7D%20f%28x%29%20dx." \* MERGEFORMATINET  Integrands with Infinite Discontinuities This is another type of integral that arises when the integrand has a vertical asymptote (an infinite discontinuity) at the limit of integration or at some point in the interval of integration. Recall from Chapter 5 in the Lesson on Definite Integrals that in order for the function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET to be integrable, it must be bounded on the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Ba%2C%20b%5D." \* MERGEFORMATINET Otherwise, the function is not integrable and thus does not exist. For example, the integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_0%5E4%20%5Cfrac%7Bdx%7D%20%7Bx%20-%201%7D" \* MERGEFORMATINET  develops an infinite discontinuity at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET because the integrand approaches infinity at this point. However, it is continuous on the two intervals  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5B0%2C%201%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%204%5D." \* MERGEFORMATINET Looking at the integral more carefully, we may split the interval  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5B0%2C4%5D%20%5Crightarrow%20%5B0%2C1%29%20%5Ccup%20%281%2C4%5D" \* MERGEFORMATINET and integrate between those two intervals to see if the integral converges.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint_0%5E4%20%5Cfrac%7Bdx%7D%20%7Bx%20-%201%7D%20%3D%20%5Cint_0%5E1%20%5Cfrac%7Bdx%7D%20%7Bx%20-%201%7D%20%2B%20%5Cint_1%5E4%20%5Cfrac%7Bdx%7D%20%7Bx%20-%201%7D." \* MERGEFORMATINET  We next evaluate each improper integral. Integrating the first integral on the right hand side,  The integral diverges because  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%28-1%29" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%280%29" \* MERGEFORMATINET are not defined, and thus there is no reason to evaluate the second integral. We conclude that the original integral diverges and has no finite value. Example 4: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_1%5E3%20%5Cfrac%7Bdx%7D%20%7B%5Csqrt%7Bx%20-%201%7D%7D" \* MERGEFORMATINET . Solution:  So the integral converges to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2%5Csqrt%7B2%7D" \* MERGEFORMATINET . Example 5: In Chapter 5 you learned to find the volume of a solid by revolving a curve. Let the curve be  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20xe%5E%7B-x%7D%2C%200%20%5Cle%20x%20%5Cle%20%5Cinfty" \* MERGEFORMATINET and revolving about the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x-" \* MERGEFORMATINET axis. What is the volume of revolution? Solution:  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123844" \* MERGEFORMATINET  From the figure above, the area of the region to be revolved is given by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%20%5Cpi%20y%5E2%20%3D%20%5Cpi%20x%5E2%20e%5E%7B-2x%7D" \* MERGEFORMATINET . Thus the volume of the solid is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=V%20%3D%20%5Cpi%20%5Cint_0%5E%7B%5Cinfty%7D%20x%5E2%20e%5E%7B-2x%7D%20dx%20%3D%20%5Cpi%20%5Clim_%7Bl%20%5Cto%20%5Cinfty%7D%20%5Cint_0%5El%20x%5E2%20e%5E%7B-2x%7D%20dx." \* MERGEFORMATINET  As you can see, we need to integrate by parts twice:  Thus  At this stage, we take the limit as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=l" \* MERGEFORMATINET  approaches infinity. Notice that the when you substitute infinity into the function, the denominator of the expression  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2l%5E2%20%2B%202l%20%2B%201%7D%20%7B-4e%5E%7B2l%7D%7D%2C" \* MERGEFORMATINET being an exponential function, will approach infinity at a much faster rate than will the numerator. Thus this expression will approach zero at infinity. Hence  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=V%20%3D%20%5Cpi%20%5Cleft%20%5B0%20%2B%20%5Cfrac%7B1%7D%20%7B4%7D%20%5Cright%5D%20%3D%20%5Cfrac%7B%5Cpi%7D%20%7B4%7D%2C" \* MERGEFORMATINET  So the volume of the solid is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cpi/4." \* MERGEFORMATINET  Example 6: Evaluate  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B-%20%5Cinfty%7D%5E%7B%2B%20%5Cinfty%7D%20%5Cfrac%20%7Bdx%7D%7Be%5Ex%2Be%5E%7B-x%7D%7D" \* MERGEFORMATINET . Solution: This can be a tough integral! To simplify, rewrite the integrand as  Substitute into the integral:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Cint%20%5Cfrac%20%7Bdx%7D%7Be%5Ex%2Be%5E%7B-x%7D%7D%20%3D%20%5Cint%20%5Cfrac%20%7Be%5Ex%7D%7B1%2B%28e%5Ex%29%5E2%7Ddx." \* MERGEFORMATINET  Using  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u-" \* MERGEFORMATINET substitution, let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20e%5Ex%2C%20du%20%3D%20e%5Exdx." \* MERGEFORMATINET   Returning to our integral with infinite limits, we split it into two regions. Choose as the split point the convenient  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET   Taking each integral separately,  Similarly,  Thus the integral converges to  Multimedia Links For a video presentation of Improper Integrals (22.0), see  HYPERLINK "http://www.youtube.com/watch?v=85-HNJyuyrU" Improper Integrals, www.justmathtutoring.com (6:23) INCLUDEPICTURE "http://i1.ytimg.com/vi/85-HNJyuyrU/hqdefault.jpg" \* MERGEFORMATINET . For a video presentation of Improper Integrals with Infinity in the Upper and Lower Limits (22.0), see  HYPERLINK "http://www.youtube.com/watch?v=f6cGotvktxs" Improper Integrals, www.justmathtutoring.com (7:55) INCLUDEPICTURE "http://i3.ytimg.com/vi/f6cGotvktxs/hqdefault.jpg" \* MERGEFORMATINET . Review Questions Determine whether the following integrals are improper. If so, explain why.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B7%7D%20%5Cfrac%20%7Bx%2B2%7D%7Bx-3%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B7%7D%20%5Cfrac%20%7Bx%2B2%7D%7Bx%2B3%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%20%7B%7B%20%5Cln%7Dx%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%20%7B1%7D%7B%5Csqrt%20%7Bx-2%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B%7B%5Cpi%7D/4%7D%20%7B%5Ctan%20x%7D%20dx" \* MERGEFORMATINET  Evaluate the integral or state that it diverges.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%20%7B1%7D%7Bx%5E%7B2.001%7D%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B-%20%5Cinfty%7D%5E%7B-2%7D%20%5Cleft%20%5B%20%5Cfrac%7B1%7D%7Bx-1%7D%20-%20%5Cfrac%20%7B1%7D%7Bx%2B1%7D%20%5Cright%20%5D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B-%5Cinfty%7D%5E%7B0%7D%20e%5E%7B5x%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B3%7D%5E%7B5%7D%20%5Cfrac%20%7B1%7D%7B%28x-3%29%5E4%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B-%7B%5Cpi%7D/2%7D%5E%7B%7B%5Cpi%7D/2%7D%20%7B%5Ctan%20x%7D%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%20%7B1%7D%7B%5Csqrt%20%7B1-x%5E2%7D%7D%20dx" \* MERGEFORMATINET  The region between the x-axis and the curve  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20e%5E%7B-x%7D" \* MERGEFORMATINET  for x ( 0 is revolved about the x-axis. Find the volume of revolution, V. Find the surface area of the volume generated, S. Review Answers Improper; infinite discontinuity at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%203." \* MERGEFORMATINET  Not improper. Improper; infinite discontinuity at  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200." \* MERGEFORMATINET  Improper; infinite interval of integration. Not improper.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B1.001%7D" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%203" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B5%7D" \* MERGEFORMATINET  divergent divergent  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B%5Cpi%7D%7B2%7D." \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=V%20%3D%20%20%5Cpi%20/2" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=S%20%3D%20%7B%5Cpi%7D%20%5Cleft%20%5B%7B%5Csqrt%20%7B2%7D%20%20%2B%20%7B%20%5Cln%7D%281%2B%20%5Csqrt%20%7B2%7D%29%7D%20%5Cright%20%5D." \* MERGEFORMATINET  Homework Evaluate the following integrals. 1.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%7B%5Csqrt%20%7B%5Csin%20x%7D%7D%20%7B%5Ccos%20x%7D%20dx" \* MERGEFORMATINET  2.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20x%20%5Ctan%5E2%28x%5E2%29%20%5Csec%5E2%28x%5E2%29%20dx" \* MERGEFORMATINET  3.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B%7B%20%5Cln%7D3%7D%20%7B%5Csqrt%20%7Be%5E%7B2x%7D%20-%201%7D%7D%20%20dx" \* MERGEFORMATINET  4.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%20%7B1%7D%7Bx%5E2%7D%20dx" \* MERGEFORMATINET  5.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B-1%7D%5E%7B8%7D%20%5Cfrac%20%7B1%7D%7B%5Csqrt%5B3%5D%7Bx%7D%7D%20dx" \* MERGEFORMATINET  6.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint%20%5Cfrac%20%7Bx%5E2%2Bx-16%7D%7B%28x%2B1%29%28x-3%29%5E2%7D%20dx" \* MERGEFORMATINET  7. Graph and find the volume of the region enclosed by the x-axis, the y-axis,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%202" \* MERGEFORMATINET  and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20x%5E2/%289-x%5E2%29" \* MERGEFORMATINET  when revolved about the x-axis. 8. The Gamma Function,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5CGamma%20%28x%29" \* MERGEFORMATINET , is an improper integral that appears frequently in quantum physics. It is defined as  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5CGamma%20%28x%29%20%3D%20%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20%7Bt%5E%7Bx-1%7De%5E%7B-t%7D%7D%20dt." \* MERGEFORMATINET  The integral converges for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cge%200." \* MERGEFORMATINET  a. Find  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5CGamma%20%281%29." \* MERGEFORMATINET  b. Prove that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5CGamma%20%28x%20%2B%201%29%20%3D%20x%20%5CGamma%28x%29" \* MERGEFORMATINET , for all  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%5Cge%200" \* MERGEFORMATINET . c. Prove that  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5CGamma%20%5Cleft%20%28%20%5Cfrac%7B1%7D%7B2%7D%20%5Cright%20%29%20%3D%20%7B%5Csqrt%20%7B%5Cpi%7D%7D." \* MERGEFORMATINET  9. Refer to the Gamma Function defined in the previous exercise to prove that (a)  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B%5Cinfty%7D%20e%5E%7B-x%5En%7D%20dx%20%3D%20%5CGamma%20%5Cleft%20%28%20%5Cfrac%7Bn%2B1%7D%7Bn%7D%20%5Cright%20%29%2C%20%7Bn%20%5Cge%200%7D" \* MERGEFORMATINET  [Hint: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%20x%5En" \* MERGEFORMATINET ] (b)  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cint_%7B0%7D%5E%7B1%7D%20%28%7B%20%5Cln%7Dx%29%5En%20%20dx%20%3D%20%7B%28-1%29%5En%7D%20%5CGamma%20%28n%2B1%29%2C%20%20%20%7Bn%20%5Cge%200%7D" \* MERGEFORMATINET  [Hint: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%20-%20%5Cln%20x" \* MERGEFORMATINET ] 10. In wave mechanics, a saw tooth wave is described by the integral  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%20%5Cint_%7B-%20%5Cpi%20/%20%5Comega%7D%5E%7B%2B%20%5Cpi%20/%20%5Comega%7D%20%7Bt%20%5Csin%28k%20%5Comega%20t%29%7D%20dt%2C" \* MERGEFORMATINET  where k is called the wave number, w is the frequency, and t is the time variable. Evaluate the integral. Answers 1.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B3%7D%7B2%7D%20%5Csin%5E%7B2/3%7D%20x%20%2B%20C" \* MERGEFORMATINET  2.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B1%7D%7B6%7D%20%5Ctan%5E3%20%28x%5E2%29%20%2B%20C" \* MERGEFORMATINET  3.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%5Cln%203%20" \* MERGEFORMATINET  4. divergent 5.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B9%7D%7B8%7D" \* MERGEFORMATINET  6.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7B%20%5Cln%7D%5Cfrac%20%7B%28x-3%29%5E2%7D%7B%5Cbegin%7Bvmatrix%7D%7Bx%2B1%7D%20%5Cend%7Bvmatrix%7D%7D%20%2B%5Cfrac%20%7B1%7D%7Bx-3%7D%20%2B%20C" \* MERGEFORMATINET  7.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7B%5Cpi%7D%20%5Cleft%20%28%20%5Cfrac%7B19%7D%7B5%7D%20-%20%5Cfrac%20%7B9%7D%7B4%7D%20%7B%20%5Cln%7D5%20%5Cright%29" \* MERGEFORMATINET  8. a.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%20%20%5CGamma%20%281%29%20%3D%201" \* MERGEFORMATINET  9. a. Hint: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%20x%5En." \* MERGEFORMATINET  b. Hint:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%20-%5Cln%20x" \* MERGEFORMATINET  10.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%20%7B2%7D%7B%28k%20%5Comega%29%5E2%7D%20%5Csin%20%28k%20%5Cpi%29" \* MERGEFORMATINET  10.7 Ordinary Differential Equations General and Particular Solutions Differential equations appear in almost every area of daily life including science, business, and many others. We will only consider ordinary differential equations (ODE). An ODE is a relation on a function  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET of one independent variable  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and the derivatives of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET with respect to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET , i.e.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%5E%7B%28n%29%7D%20%3D%20F%28x%2C%20y%2C%20%20y%27%20%2C%20%5Cldots%20.%2Cy%5E%7B%28n%20-%201%29%7D%29" \* MERGEFORMATINET . For example,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%27%27%20%2B%20%28y%27%20%29%5E2%20%2B%20y%20%3D%20x" \* MERGEFORMATINET . An ODE is linear if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F" \* MERGEFORMATINET can be written as a linear combination of the derivatives of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET , i.e.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%5E%7B%28n%29%7D%20%3D%20%5Csum%20a_i%28x%29y%28i%29%20%2B%20r%28x%29" \* MERGEFORMATINET . A linear ODE is homogeneous if  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=r%28x%29%20%3D%200" \* MERGEFORMATINET . A general solution to a linear ODE is a solution containing a number (the order of the ODE) of arbitrary variables corresponding to the constants of integration. A particular solution is derived from the general solution by setting the constants to particular values. For example, for linear ODE of second degree  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%27%27%20%2B%20y%20%3D%200" \* MERGEFORMATINET , a general solution have the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y_g%20%3D%20A%20%5Ccos%20x%20%2B%20B%20%5Csin%20x" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%2C%20B" \* MERGEFORMATINET are real numbers. By setting  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%201" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=B%20%3D%200%2C%20y_p%20%3D%20%5Ccos%20x" \* MERGEFORMATINET  It is generally hard to find the solution of differential equations. Graphically and numerical methods are often used. In some cases, analytical method works, and in the best case,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET has an explicit formula in  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET . Multimedia Links For a video introduction to differential equations (27.0), see  HYPERLINK "http://www.youtube.com/watch?v=UUCN68SIDNI" Math Video Tutorials by James Sousa, Introduction to Differential Equations (8:12) INCLUDEPICTURE "http://i2.ytimg.com/vi/UUCN68SIDNI/hqdefault.jpg" \* MERGEFORMATINET . Slope Fields and Isoclines We now only consider linear ODE of the first degree, i.e.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20F%28x%2C%20y%29" \* MERGEFORMATINET . In general, the solutions of a differential equation could be visualized before trying an analytic method. A solution curve is the curve that represents a solution (in the  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=xy%20-" \* MERGEFORMATINET plane). The slope field of the differential equation is the set of all short line segments through each point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28x%2C%20y%29" \* MERGEFORMATINET and with slope  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%2C%20y%29" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123846" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123843" \* MERGEFORMATINET  An isocline (for constant  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=k" \* MERGEFORMATINET ) is the line along which the solution curves have the same gradient  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28k%29" \* MERGEFORMATINET . By calculating this gradient for each isocline, the slope field can be visualized; making it relatively easy to sketch approximate solution curves. For example, INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7Bx%7D%7By%7D" \* MERGEFORMATINET . The isoclines are  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%5Cfrac%7Bx%7D%7Bk%7D" \* MERGEFORMATINET . Example 1 Consider  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7Bx%7D%7By%5E2%7D" \* MERGEFORMATINET . We briefly sketch the slope field as above. The solutions are  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%5E3%20%3D%5Cfrac%7B3%7D%7B2%7D%5C%20x%5E2%2BC" \* MERGEFORMATINET .  INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123849" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/images?id=123848" \* MERGEFORMATINET  Exercise Sketch the slope field of the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D1-y" \* MERGEFORMATINET . Sketch the solution curves based on it. Sketch the slope field of the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3Dy-x" \* MERGEFORMATINET . Find the isoclines and sketch a solution curve that passes through  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%281%2C%200%29" \* MERGEFORMATINET . Differential Equations and Integration We begin the analytic solutions of differential equations with a simple type where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%2Cy%29" \* MERGEFORMATINET is a function of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET only.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20f%28x%29" \* MERGEFORMATINET is a function of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET . Then any antiderivative of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=f" \* MERGEFORMATINET is a solution by the Fundamental Theorem of Calculus:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bd%7D%7Bdx%7D%5Cint_%7Ba%7D%5E%7Bx%7D%5C%20f%28t%29%5C%20dt%3Df%28x%29" \* MERGEFORMATINET . Example 1 Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20x" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%280%29%20%3D%201" \* MERGEFORMATINET . Solution.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%5Cint%5C%20x%5C%20dx%20%3D%5Cfrac%7Bx%5E2%7D%7B2%7D%2BC" \* MERGEFORMATINET . Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%280%29%20%3D%201" \* MERGEFORMATINET gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1%20%3D%200%20%2B%20C" \* MERGEFORMATINET , i.e.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C%20%3D%201" \* MERGEFORMATINET Therefore  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%5Cfrac%7Bx%5E2%7D%7B2%7D%2B%201" \* MERGEFORMATINET . Example 2 Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D" \* MERGEFORMATINET . Solution. We have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%5Cint%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1%20%2B%20x%5E2%7D%7D%5C%20dx" \* MERGEFORMATINET and a substitution  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20%5Csinh%5E%7B-1%7D%20x" \* MERGEFORMATINET gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%3D%5Cint%20%5Cfrac%7B%5Ccos%20h%5C%20u%5C%20du%7D%7B%5Csqrt%7B1%2B%20%5Csin%20h%5E2%5C%20u%7D%7D%3D%5Cint%5Cfrac%7B%5Ccos%20h%5C%20u%5C%20du%7D%7B%5Ccos%20h%5C%20u%7D%3D%5Cint%5C%20du%3Du%3D%5Csin%20h%5E%7B-1%7Dx%2BC" \* MERGEFORMATINET . Exercise Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20%5Csqrt%7B9-x%5E2%7D" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%280%29%20%3D%203" \* MERGEFORMATINET . Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B1%2Bx%2Bx%5E2%7D%7D" \* MERGEFORMATINET . Hint: Let  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=u%20%3D%20x%20%2B%5Cfrac%7B1%7D%7B2%7D" \* MERGEFORMATINET . Solving Separable First-Order Differential Equations The next type of differential equation where analytic solution are rela- tively easy is when the dependence of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%2C%20y%29" \* MERGEFORMATINET on  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET are separable:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%20F%28x%2C%20y%29" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=F%28x%2C%20y%29%20%3D%20f%28x%29%20g%28y%29" \* MERGEFORMATINET is the product of a functions of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET respectively. The solution is in the form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%28x%29%20%3D%20Q%28y%29" \* MERGEFORMATINET . Here  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28y%29" \* MERGEFORMATINET is never  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0" \* MERGEFORMATINET or the values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET in the solutions will be restricted by where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=g%28y%29%20%3D%200" \* MERGEFORMATINET . Example 1 Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%27%20%3D%20xy" \* MERGEFORMATINET with the initial condition  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%280%29%20%3D%201" \* MERGEFORMATINET . Solution. Separating  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET turns the equation in differential form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%7By%7D%3Dxdx" \* MERGEFORMATINET . Integrating both sides, we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%7Cy%7C%20%3D%5Cfrac%7B1%7D%7B2%7Dx%5E2%20%2B%20C" \* MERGEFORMATINET . Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%280%29%20%3D%201" \* MERGEFORMATINET gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%7C1%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7D%280%29%5E2%20%2B%20C" \* MERGEFORMATINET , i.e.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=C%20%3D%200" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%7Cy%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7Dx%5E2" \* MERGEFORMATINET . So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%7Cy%7C%20%3D%20e%5E%7B%5Cfrac%7B1%7D%7B2%7Dx%5E2%7D." \* MERGEFORMATINET  Therefore, the solutions are  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20%5Cpm%20e%5E%7B%5Cfrac%7B1%7D%20%7B2%7Dx%5E2%7D" \* MERGEFORMATINET . Here  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=Q%28y%29%20%3D%20y" \* MERGEFORMATINET is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0" \* MERGEFORMATINET when  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%200" \* MERGEFORMATINET and the values of  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET in the solutions satisfy  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3E%200" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3C%200" \* MERGEFORMATINET . Example 2. Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2xy%27%20%3D%201%20-%20y%5E2" \* MERGEFORMATINET . Solution. Separating  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y" \* MERGEFORMATINET turns the equation in differential form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2%7D%20%7B1%20-%20y%5E2%7D%20dy%20%3D%20%5Cfrac%7Bdx%7D%20%7Bx%7D." \* MERGEFORMATINET  Resolving the partial fraction  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B2%7D%20%7B1%20-%20y%5E2%7D%20%3D%20%5Cfrac%7BA%7D%20%7B1%20-%20y%7D%20%2B%20%5Cfrac%7BB%7D%20%7B1%20%2B%20y%7D" \* MERGEFORMATINET gives linear equations  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%2B%20B%20%3D%202" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20-%20B%20%3D%200" \* MERGEFORMATINET . So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%5Cfrac%7B1%7D%20%7B1%20-%20y%7D%20%2B%20%5Cfrac%7B1%7D%20%7B1%20%2B%20y%7D%5Cright%20%29%20dy%20%3D%20%5Cfrac%7Bdx%7D%20%7Bx%7D" \* MERGEFORMATINET . Integrating both sides, we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=-%5Cln%20%7C%201%20-%20y%20%7C%20%2B%20%5Cln%20%7C%201%20%2B%20y%20%7C%20%3D%20%5Cln%20%7C%20x%20%7C%20%2B%20C" \* MERGEFORMATINET or  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cln%20%5Cleft%20%7C%5Cfrac%7B1%20%2B%20y%7D%20%7B1%20-%20y%7D%5Cright%20%7C%20%3D%20%5Cln%20%28e%5EC%20%7C%20x%20%7C%20%29%20%3D%20%5Cln%20D%20%7C%20x%20%7C" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%20%3D%20e%5EC%20%3E%200" \* MERGEFORMATINET . Then  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%7C%5Cfrac%7B1%20%2B%20y%7D%20%7B1%20-%20y%7D%5Cright%20%7C%20%3D%20D%20%7C%20x%20%7C" \* MERGEFORMATINET , i.e.  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B1%20%2B%20y%7D%20%7B1%20-%20y%7D%20%3D%20%5Cpm%20Dx" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%20%3E%200" \* MERGEFORMATINET . Therefore, the solution has form  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20%5Cpm%20%5Cfrac%7BDx%20%2B%201%7D%20%7BDx%20-%201%7D" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=D%20%3E%200" \* MERGEFORMATINET . Exercise Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%20%7Bdx%7D%20%3D%20%5Cfrac%7B1%7D%20%7Be%5Ey%7D" \* MERGEFORMATINET which satisfies the condition  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%28e%29%20%3D%200" \* MERGEFORMATINET . Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%20%7Bdx%7D%20%3D%20x%28y%5E2%20%2B%201%29" \* MERGEFORMATINET . Solve the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%20%7Bdx%7D%20%3D%20%5Cfrac%7Bx%7D%20%7B%5Csqrt%7B1%20-%20y%5E2%7D%7D" \* MERGEFORMATINET . Exponential and Logistic Growth In some model, the population grows at a rate proportional to the current population without restrictions. The population is given by the differential equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7BdP%7D%20%7Bdt%7D%20%3D%20kP" \* MERGEFORMATINET , where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=k%20%3E%200" \* MERGEFORMATINET are growth. In a refined model, the rate of growth is adjusted by another factor  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%281%20-%20%5Cfrac%7BP%7D%20%7BK%7D%5Cright%20%29" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=K" \* MERGEFORMATINET is the carrier capacity. This is close to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=1" \* MERGEFORMATINET when  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P" \* MERGEFORMATINET is small compared with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=K" \* MERGEFORMATINET but close to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=0" \* MERGEFORMATINET when  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P" \* MERGEFORMATINET is close to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=K" \* MERGEFORMATINET . Both differential equations are separable and could be solved as in last section. The solutions are respectively:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%28t%29%20%3D%20P%280%29e%5E%7Bkt%7D" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%28t%29%20%3D%20%5Cfrac%7BP_0%7D%20%7B1%20%2B%20Ae%5E%7Bkt%7D%7D" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%20%5Cfrac%7BK%20-%20P_0%7D%20%7BP_0%7D" \* MERGEFORMATINET . Example 1 (Exponential Growth) The population of a group of immigrant increased from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=10000" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=20000" \* MERGEFORMATINET from the end of first year to the end of second year they came to an island. Assuming an exponential growth model on the population, estimate the size of the group of initial immigrants. Solution. The population of the group is given by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%20P_%7B0%7De%5E%7Bkt%7D" \* MERGEFORMATINET where the initial population and relative growth rate are to be determined. At  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%201" \* MERGEFORMATINET (year),  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%2010000" \* MERGEFORMATINET , so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=10000%20%3D%20P_%7B0%7De%5E%7Bk%20%5Ccdot%201%7D%20%3D%20P_%7B0%7D%5E%7Bek%7D" \* MERGEFORMATINET . At  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%202" \* MERGEFORMATINET (year),  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%2020000" \* MERGEFORMATINET , so  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=20000%20%3D%20P_%7B0%7De%5E%7Bk%20%5Ccdot%202%7D%20%3D%20P_%7B0%7De%5E%7B2k%7D" \* MERGEFORMATINET . Dividing both sides of the second equation by the first, we have  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=2%20%3D%20e%5E%7Bk%7D" \* MERGEFORMATINET . Then back in the first equation,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=10000%20%3D%20P_0%282%29" \* MERGEFORMATINET . So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P_0%20%3D%205000" \* MERGEFORMATINET . There are  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=5000" \* MERGEFORMATINET initial immigrants. Example 2 (Logistic Growth) The population on an island is given by the equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7BdP%7D%20%7Bdt%7D%20%3D%200.05P%5Cleft%20%281%20-%20%5Cfrac%7BP_0%7D%20%7B5000%7D%5Cright%20%29%2C%20P_0%20%3D%201000" \* MERGEFORMATINET . Find the population sizes  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%2820%29%2C%20P%2830%29" \* MERGEFORMATINET . At what time will the population first exceed  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=4000" \* MERGEFORMATINET ? Solution. The solution is given by  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=P%20%3D%20%5Cfrac%7BP_0%7D%20%7B1%20-%20Ae%5E%7B0.05t%7D%7D" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=A%20%3D%20%5Cfrac%7B5000%20-%201000%7D%20%7B1000%7D%20%3D%204" \* MERGEFORMATINET .  Solve for time,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=4000%20%3D%20%5Cfrac%7B5000%7D%20%7B1%20%2B%204e%5E%7B-0.05%28t%29%7D%7D" \* MERGEFORMATINET gives  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=e%5E%7B%28-0.05t%29%7D%20%3D%20%5Cfrac%7B%5Cfrac%7B5000%7D%20%7B4000%7D%20-%201%7D%20%7B4%7D%20%3D%200.0625" \* MERGEFORMATINET . So  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=t%20%3D%2056" \* MERGEFORMATINET . The population first exceed 4000 in the 56th year. Exercise (Exponential Growth) The population of a suburban city increased from 10,000 in 2005 to 30,000 in 2007. Assuming an exponential growth model on the population, by which year will the population first exceeds 100,000? (Logistic Growth) The population of a city is given by the equation  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7BdP%7D%20%7Bdt%7D%20%3D%200.06P%5Cleft%20%281%20-%20%5Cfrac%7BP_0%7D%20%7B100000%7D%5Cright%20%29%2C%20P_0%20%3D%2025000" \* MERGEFORMATINET . Find the population sizes P(10) and P(25). At what time will the population first exceed 90,000? Multimedia Links For a video presentation of Differential Equations including growth and decay (27.0), see  HYPERLINK "http://www.youtube.com/watch?v=GtBClD5m7Cs" Differential Equations, Growth and Decay (7:23) INCLUDEPICTURE "http://i4.ytimg.com/vi/GtBClD5m7Cs/hqdefault.jpg" \* MERGEFORMATINET . Numerical Methods (Euler's, Improved Euler, Runge-Kutta) The Euler's method is a numerical approximation to a solution curve starting from the point  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%28a%2C%20b%29" \* MERGEFORMATINET through the algorithm:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y_%7Bn%20%2B%201%7D%20%3D%20y_%7Bn%7D%20%2B%20hF%28x_%7Bn%7D%2C%20y_%7Bn%7D%29" \* MERGEFORMATINET where  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x_%7B0%7D%20%3D%20a%2C%20y_%7B0%7D%20%3D%20b" \* MERGEFORMATINET and  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=h" \* MERGEFORMATINET is the step size. The shorter step size, the better is the approximation to the solution curve. Improved Euler (Heun) method adapts on Euler's method by using both end point values:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y_%7Bn%2B1%7D%20%3D%20y_n%20%2B%20%5Cfrac%7Bh%7D%20%7B2%7D%20%5BF%28x_n%2C%20y_n%29%20%2B%20F%28x_%7Bn%20%2B%201%7D%2C%20y_%7Bn%20%2B%201%7D%29%5D." \* MERGEFORMATINET  Since  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y_%7Bn%2B1%7D" \* MERGEFORMATINET also appears on the right side, we replace it by Euler's formula,  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=y_%7Bn%2B1%7D%20%3D%20y_n%20%2B%20%5Cfrac%7Bh%7D%20%7B2%7D%20%5BF%28x_n%2C%20y_n%29%20%2B%20F%28x_%7Bn%20%2B%201%7D%2C%20y_n%20%2B%20hF%28x_n%2C%20y_n%29%29%5D." \* MERGEFORMATINET  The Runge-Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of our ODE. On them, apply Simpson's rule:  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=y_%7Bn%20%2B%201%7D%20-%20y_n%20%3D%20%5Cint_%7Bx_n%7D%5E%7Bx_%7Bn%20%2B%201%7D%7D%20f%27%28x%29%20dx%20%3D%20%5Cint_%7Bx_n%7D%5E%7Bx_n%20%2B%20h%7D%20f%27%28x%29%20dx" \* MERGEFORMATINET   INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?blockmath=%5Capprox%20%5Cfrac%7Bh%7D%20%7B6%7D%20%5Cleft%20%5C%7By%27%28x_n%29%20%2B%204y%27%5Cleft%20%28x_n%20%2B%20%5Cfrac%7Bh%7D%20%7B2%7D%5Cright%20%29%20%2B%20y%27%20%28x_n%20%2B%201%29%5Cright%20%5C%7D." \* MERGEFORMATINET  Exercise 1. Apply the Euler's, improved Euler's and the Runge-Kutta methods on the ODE  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7Bdy%7D%20%7Bdx%7D%20%3D%20y" \* MERGEFORMATINET to approximate the solution that satisfy  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%280%29%20%3D%201" \* MERGEFORMATINET from  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%200" \* MERGEFORMATINET to  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=x%20%3D%201" \* MERGEFORMATINET with  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=h%20%3D%200.2" \* MERGEFORMATINET . We know the exact solution is  INCLUDEPICTURE "http://www.ck12.org/ck12/ucs/?math=y%20%3D%20e%5Ex" \* MERGEFORMATINET . Compare their relative accuracy against the exact solution. Texas Instruments Resources In the CK-12 Texas Instruments Calculus FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See  HYPERLINK "http://www.ck12.org/flexr/chapter/9732"  HYPERLINK "http://www.ck12.org/flexr/chapter/9732" http://www.ck12.org/flexr/chapter/9732.     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