ࡱ> *,)5@ bjbj22 @XXT::::\4d32bbbb2222222$4R6v 3" 3bb3uuubb2u2uu02b Ji%:<12430d31^7f^7(2^72@u 3 3dVk VIntegration By Parts Overview: In this lesson, students will be introduced to another method of integration, integration by parts. While substitution helps us recognize derivatives produced by the chain rule, integration by parts helps us to recognize derivatives produced by the product rule. Grade Level/Subject: The lesson is for 12th graders in AP Calculus. Time: 1-50 minute class period Purpose: This lesson will give students yet another method of solving complicated integrals. It will also give them a small introduction into differential equations. Prerequisite Knowledge: Student should: Understand Definite Integrals Understand and be able to perform Integration by Substitution Know the product rule for derivatives Objectives: Students will learn to do integration by parts and be able to use the method comfortably. Students will learn how do decide when to use integration by substitution and when to use integration by parts. Standards: Problem-Solving: Students will use their problem-solving techniques to determine which method to use for particular integrals. Communications: Students will have to communicate clearly what method is being used and all the steps taken to achieve their answer. Connections: Students will need to recall the product rule from differential calculus to recognize integrals that can be solved through integration by parts. Resources/Materials Needed: Calculus Book Dry Erase Board/Dry Erase Markers Attached Handouts and Worksheets Activities and Procedures: Begin by having the students recall the product rule from differential calculus.  EMBED Equation.3  If we write u for f(x) and v for g(x), we have  EMBED Equation.3  From here, we can reverse the product rule:  EMBED Equation.3  When we subtract  EMBED Equation.3  from both sides, we get the integration by parts formula.  EMBED Equation.3  Give the steps for using Integration by Parts. Step 1: Let u = f(x) and dv = g(x)dx, where f(x)g(x)dx is the original integrand. Good choices to make are integrals  EMBED Equation.3 , which are easy to integrate. Step 2: Compute du = f(x)dx and v =  EMBED Equation.3 . Step 3: Substitute u, v, du, and dv into the formula,  EMBED Equation.3  Step 4: Calculate uv -  EMBED Equation.3  If  EMBED Equation.3  is difficult or impossible to integrate, go back to step 1 and consider other choices for u and dv. Step 5: Check your solution by differentiating and comparing to the original integrand. Do several examples to demonstrate the method. Have the students do one within their groups and one on their own. The best way to learn this method is to practice, practice, practice!  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Give the two attached Handouts to the students to help them. Homework: Read Section 6.9 about improper integrals and take notes. Complete the attached Worksheet.  1. Let u = f(x) and dv = g(x)dx, where f (x)g(x)dx is the original integrand. Try making dv be the most complicated portion of the integrand that fits a basic integration formula. Then u will be the remaining factor(s) of the integrand. Try letting u be the portion of the integrand whose derivative is a simpler function and then dv will be the remaining factor(s) of the integrand. 2. Compute du = f (x) dx and  EMBED Equation.DSMT4 . 3. Substitute u, v, du, and dv into the formula  EMBED Equation.DSMT4  4. Evaluate  EMBED Equation.DSMT4 . (If the integral is difficult or impossible to integrate, go back to Step 1 and consider other choices for u and dv. 5. Check your solution by differentiating and comparing it to the original integrand. Summary of Common Integrals using Integration by Parts 1. For integrals of the form  EMBED Equation.DSMT4  let u = xn let dv = eax or sin(ax)dx, or cos(ax)dx 2. For integrals of the form  EMBED Equation.DSMT4 let u = ln(x), arcsin(ax), or arctan(x) let dv = xndx 3. 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