ࡱ> XZ Wq` a&bjbjqPqP .::PjjjjF4F4F48~445ohZ6<(8<8< ?LPdQGoIoIoIoIoIoIo$0qhsZmoRIL"RRmojj8< ? o(j(j(jR|jZ8< ?Go(jRGo(j(jl@Sn ?N6 ФMF4*e'nsno0o=ns@hs,SnsSn RR(jRRRRRmomoi^RRRoRRRR,F4F4jjjjjj INTEGRAL  1. ANTI TURUNAN Definisi  Contoh : F(x) = cos x anti turunan dari f (x) = (sin x sebab F(x) = (sin x a(x) = 2x2 anti turunan dari f (x) = 4x sebab a (x) = 4x v(x) =  EMBED Equation.2  x 3 anti turunan dari g(x) = x2 sebab v (x) = x2 Definisi  Definisi  Bentuk  EMBED Equation.2 f (x) dx dinamakan integral tak tentu dari fungsi y = f (x) Lambang  EMBED Equation.2  dinamakan integral yaitu merupakan operasi anti differensial Dalil 1  Dalil 2  Contoh : 1. Hitung  EMBED Equation.2  Jawab :  EMBED Equation.2  =  EMBED Equation.2  =  EMBED Equation.2  + 5x + C 2. Tentukan  EMBED Equation.2  Jawab :  EMBED Equation.2  = EMBED Equation.2  =  EMBED Equation.3  3. Tentukan  EMBED Equation.2  Jawab :  EMBED Equation.2  =  EMBED Equation.2  =  EMBED Equation.2  =  EMBED Equation.2  4. Tentukan  EMBED Equation.2  Jawab :  EMBED Equation.2  =  EMBED Equation.2  =  EMBED Equation.2  =  EMBED Equation.2  5. Tentukan  EMBED Equation.2 dx Jawab :  EMBED Equation.2 dx =  EMBED Equation.2 dx =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  6. Gradien garis singgung pada grafik y = (x) di setiap titik (x , y) dinyatakan oleh bentuk dy/dx = 2x ( 5. Bila grafik y = f (x) melalui titik A (1 , 7), tentukan persamaan fungsi y = f (x) ! Jawab :  EMBED Equation.2  = 2x ( 5 ( dy = (2x - 5) dx ( dy = (2x ( 5) dx ( y =  EMBED Equation.2  = x2 ( 5x + C Grafik melalui titik A(1 , 7), jadi 7 = 12 ( 5(1) + C didapat C = 11 Akibatnya persamaan y = f (x) adalah y = f (x) = x2 ( 5x + 11 2. INTEGRAL DENGAN SUBSTITUSI Misalkan u = u (x) dan y = f (u) masing-masing anti turunan dari u'(x) dan f ' (u), maka :  Bentuk integral di atas, dikenal dengan bentuk integral dengan subtitusi. Dalil 3  Contoh : 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3 =  EMBED Equation.3  =  EMBED Equation.3  3. INTEGRAL PARSIAL Misalkan u = u(x) dan v = v(x) fungsi-fungsi yang differensiabel pada daerahnya, maka  dinamakan bentuk integral parsial. Contoh : Tentukan  EMBED Equation.3  Jawab : Misalkan u = x dan dv = sin x dx, maka didapat du=dx dan v = (cos x  EMBED Equation.3  = ( x cos x ( EMBED Equation.3  = .. dst. Tentukan  EMBED Equation.3  dengan rumus integrasi parsial Jawab : Misalkan u = x dan dv =  EMBED Equation.3  maka du = dx dan v = 2 EMBED Equation.3   EMBED Equation.3 =2x EMBED Equation.3 ( EMBED Equation.3  dst. 4. INTEGRAL TERTENTU Definisi  Bentuk integral di atas disebut integral tertentu dari y = f(x) a dan b disebut batas integral dengan a merupakan batas bawah dan b merupakan batas atas. Dalil 4  Contoh : Hitung  EMBED Equation.3  Jawab :  EMBED Equation.3 = 1/3 x3 x2 ( EMBED Equation.3  =  EMBED Equation.3  Hitung  EMBED Equation.3  Jawab :  EMBED Equation.3 = .sin (2t (() ( EMBED Equation.3  = [sin (2( ( () sin (0 ( ()] = [sin ( sin (( ()] = 0 Hitung  EMBED Equation.3  Jawab :  EMBED Equation.3  =  EMBED Equation.3  EMBED Equation.3 ( EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  5. LUAS DAERAH Misalkan y = f(x) berharga positif pada daerah a ( x ( b dan kontinu pada daerah tersebut, maka luas daerah yang dibatasi oleh grafik y = f(x) dengan sumbu x dari x = a ke x = b adalah  Bila y = f(x) berharga negatif pada daerah a ( x ( b maka luas daerah yang dibatasi oleh y = f(x) dengan semubu x dari x = a ke x = b adalah  Misalkan f(x) ( g (x) pada daerah a ( x ( b maka luas daerah yang dibatasi oleh grafik y = f(x) dan y = g(x) adalah  Tentukan luas daerah yang dibatasi oleh grafik y = (x2 + 2x dengan sumbu x Jawab : L = EMBED Equation.3  =  EMBED Equation.3 ( EMBED Equation.3  = (( EMBED Equation.3 . 8 + 4) 0 =  EMBED Equation.3  Tentukan luas daerah yang dibatasi oleh grafik y = x2 dengan garis y = x + 8 Jawab : y = x2 ... (1)  y = x + 6 (2) Dari (1) dan (2) didapat x2 = x + 6 x2 x 6 = 0 x1 = 3 ; x2 = (2 Luas daerah, L = EMBED Equation.3  = ( EMBED Equation.3 + 18 9) ( (2 12 +  EMBED Equation.3 ) = 4 + 51/3 = 21  EMBED Equation.3  6. ISI BENDA PUTAR Misalkan y = f(x) terdefinisi dan integrabel pada daerah a ( x ( b, bila daerah yang dibatasi oleh y = f(x) dan sumbu x dari x = a ke x = b diputar mengelilingi sumbu x, maka isi benda putar yang terjadi adalah :  Contoh : Tentukan isi benda putar bila daerah yang dibatasi oleh grafik y = x2 dari x = 0 ke x =1 diputar mengeliling sumbu x Jawab : Isi benda putar yang terjadi I = ( EMBED Equation.3  2. Tentukan isi benda putar bila daerah yang dibatasi oleh grafik y = x2 dan garis y = x + 2 diputar mengeliling sumbu x Jawab : Batas integral  EMBED Equation.3 ( x2 = x + 2 x2 x 2 = 0 didapat x1 = (1 dan x2 = 2. Isi benda putar yang terjadi : I= ( EMBED Equation.3  =  EMBED Equation.3 = EMBED Equation.3  LATIHAN SOAL  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Hitung luas daerah tertutup D yang dibatasi oleh parabola f(x) = 4x ( x2, garis x=1 dan sumbu X. Tunjukkan bahwa  EMBED Equation.3      Irvan Dedy, S.Pd Page  PAGE 6 of  NUMPAGES 8 SMA Dwiwarna Misalkan fungsi f terdefinisi pada selang terbuka I. Fungsi F yang memenuhi F (x) = f (x) pada I dinamakan anti turunan atau fungsi primitif dari fungsi f pada I. Anti diferensial dari fungsi f pada selang terbuka I adalah bentuk yang paling umum dari anti turunan atau fungsi primitif dari f pada selang tersebut. Jika F'(x) = f(x) pada selang terbuka I, maka anti diferensial dari fungsi f pada I adalah y = F(x) + C, C konstanta. Misalkan y = F(x) + C adalah anti turunan dari y = f (x) maka :  EMBED Equation.2 = F(x) + C 1.  EMBED Equation.2 dx = ax + C 5.  EMBED Equation.2  dx = ln x + C 2.  EMBED Equation.2 dx =  EMBED Equation.2 + C ; n ( (1 6.  EMBED Equation.2 dx =  EMBED Equation.3  + C 3.  EMBED Equation.2  dx = ( cos x + C 7.  EMBED Equation.2  dx = tg x + C 4.  EMBED Equation.2 dx = sin x + C 8.  EMBED Equation.2 dx = ( ctg x + C 1.  EMBED Equation.2 dx =  EMBED Equation.2 f (x) dx (  EMBED Equation.2 g (x) dx 2.  EMBED Equation.2 dx = k.  EMBED Equation.2 f (x) dx ; k suatu konstanta.  EMBED Equation.3   EMBED Equation.3  5.  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