10. cos (ax+b)sin(ax+b) dx = 12. sec x dx = tan x + c. 13. sec (ax+b)dx = tan (ax+b)+ c. 14. c sec x dx = - ctg x + c

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1 Integrl Tk Tentu INTEGRAL. Rumus Integrl Fungsi Aljr. k x n k n +. ( x + n ( n +. x ln x + x n + + ; n - n+ (x+ + ; dn 4. ( f ( x ± g( x f ( x ± g ( x n - n. os (x+sin(x+ ( n + n+ os (x+ + ( + (. sin x os x sin x +sin x. se x tn x +. se (x+ tn (x++ 4. se x - tg x + 5. se (x+ - tg (x++. Rumus Integrl Fungsi Trigonometri. sin x - os x + 6. tn x sex se x + 7. tn x sex -se x +. os x sin x +. tn x sin x os x - ln os x + d os x sin 4. tgx x x d os x os x 5. sin( x + - os (x+ + sin x ln sin x + sin. Integrl Prsil u dv uv - v du Didpt dri : y u.v dimn u g(x dn v h(x y u v + u v v u + u v 6. os( x + sin (x tn( x + - ln os(x+ + dy du dv v. + u. dy v du + u dv (diklikn 8. tg( x + ln sin(x+ + n 9. sin (x+ os(x+ ( n + sin n + (x+ + d (u.v v du + u dv ( u. v d v du + u dv u.v v du + u dv u dv uv - v du -

2 Integrl Tertentu f ( x F(x F( F(. Jik f(x > dn f(x < (Kurv segin erd di wh sumu x dn segin linny erd di ts sumu x. Lus Derh Antr Kurv dn Sumu- sumu Koordint Lus derh yng ditsi oleh kurv y f(x, sumu x dn gris-gris x dn x sert x g(y, sumu y dn gris-gris y dn y dpt diedkn s. Jik f(x > (Kurv di ts sumu x L - f ( x + f ( x f x ( + f ( x d. jik g(y > (kurv erd di seelh knn sumu y L f ( x. Jik f(x < (Kurv di wh sumu x L g ( y dy L - f x ( f ( x -

3 e. jik g(y < (kurv erd di seelh kiri sumu y. Lus Derh Antr Du Kurv. Di ts sumu x L - g ( y dy g ( y dy f. jik g(y < dn g(y > (kurv segin erd di seelh kiri sumu y dn segin linny erd seelh knn sumu y L y -. Di wh sumu x y ( y y i L - g ( y dy + g ( y dy g ( y dy + g ( y dy L - y - { - ( y y y } y - y -

4 . Di seelh knn sumu y L x dy -. Volume Bend Putr xdy ( x x dy. Diputr terhdp sumu x mk, V π y. Diputr terhdp sumu y mk, V π x dy - 4

5 Contoh Sol: UN UN UN. Nili dri x(x C. 56 E D. 48 Jw: x(x + 4 ( 6x + 8x x + 4x (7-(- + 4 ( Jwnny dlh A UN. Hsil dri sin x π os x π. os (x π + C D. os (x π + C - os (x π + C E. os (x π + C C. os (x π + C Jw: sin A sin A osa sin A osa sin A sin x π os x π sin x π sin( x π Jwnny dlh B os( x π +C UN π. ( sin x os x C. - Jw: sin A sin A osa π D. ( sin x os x ( sin x π {os. π os } os x E. {osπ os} { } { } Jwnny dlh E UN 4. Lus derh yng ditsi oleh kurv y 4 - x, y x, sumu Y, dn x dlh. 6 Stun lus D. Stun lus 5 Stun lus E. stun lus C. 5 Stun lus Jw: But grfikny dengn memsukkn nili x dn y : Kurv y 4 - x Jik x y 4 x y 4 - dst π - 5

6 kurv y x jik x y x y dst UN 5. Volume end putr yng terjdi jik derh yng ditsi oleh kurv yx, gris yx di kudrn I diputr 6 terhdp sumu X dlh. 64 π Stun volume D. π Stun volume π Stun volume E. π Stun volume C. π Stun volume 5 Jw: Titk potong kurv y4-x dengn gris yx 4-x x x +x 4 (x + 4 (x - x -4 tu x pd gmr terliht titik potong yng msuk dlm perhitungn dlh di x L L I + L II L I {(4 x x} 4x- x x L II (4 { x (4 x } 4( + (8 ( 4( + (7 6 x x 4x L L I + L II + 5 stun lus Titik potongny: x x x - x x(x- x tu x Volume π ( y y ( ( x π π ( 4 4 x π ( 4x x 5 5 x x Jwnny dlh C - 6

7 π ( π 5 5 π ( 8 64 π 5 Jwnny dlh D UN 6. Hsil D. 4 π ( 5 5 E. C. UN 8. Lus derh yng ditsi kurv y 4 x, y -x +, dn x dlh... Jw: stun lus C. stun lus D. stun lus stun lus E. stun lus Jw: Jwnny dlh E UN! 7. Hsil sin os... E. Jw: Integrl! sin os C. - os x + sin x! - (os π os + (sin π sin - (- + ( Jwnny dlh D D. y 4 x, y -x +, dn x titik potong 4 x -x + x -x (x (x + x tu x - x - untuk perhitungn tidk erlku kren syrt x sehingg ts ts dlh dn ts wh posisi ts dlh y 4 x dn wh dlh y -x + mk Lusny dlh "4 # Jwnny dlh B x stun lus - 7

8 UN 9. Hsil dri $%& x sin x... &'( D. $%& $%& E. &'( C. $%& Jw: misl: u os x du - sin x $%& x sin x *. * Jwnny dlh B. * + C. $%& + C UN. Volume end putr yng terjdi jik derh yng ditsi oleh kurv y x, gris y x di kudrn I diputr 6 terhdp sumu X dlh... π stun volume π stun volume C. π stun volume Jw: D. π stun volume E. π stun volume y x ; y x diputr terhdp sumu x titik potong y y x x x x x(x - x tu x Volume end putr diputr terhdp sumu x : V π y V π ( y y π (( x ( x 4 π ( 4x x π ( - π ( - π ( π ( + Jwnny dlh D π stun volume

9 UN,. Hsil,. +, C D E Jw: misl: u x + 9x du (6x du (x+ du (x+,,. +, Jwnny dlh C UN. * (. *. Lus derh yng ditsi oleh kurv y x - 4x + dn y x dlh... stun lus D.. du. + C. * + C +9 + stun lus tnp gmr : kurv y x - 4x + x 4x + >, kurv teruk ke ts sehingg gris y x erd di ts. Titik potong kurv dn gris yng merupkn tsny: msukkn persmn gris ke dlm kurv: x x - 4x + x - 4x + + x x - x x (x x dn x ts ts dn wh L ( y y ( x ( x 4x + ( x x + 4x x ( x x - x stun lus Jwnny C UN. Volume end putr yng terjdi il derh yng ditsi oleh kurv y x dn y 4x - diputr 6 mengelilingi sumu X dlh... C. π stun volume D. π stun volume π stun volume E. π stun volume π stun volume + stun lus E. stun lus C. + stun lus Jw: - 9

10 Jw: Volume end diputr terhdp sumu x mk, V π y y x >, kurv teruk ke ts gris y 4x - erd di ts kurv titik potong/ts: susitusikn persmn y 4x - ke dlm persmn y x : 4x - x x 4x + (x (x- x dn x ts ts dn wh V π y π (( 4x ( x 4 π (( 6x 4x + 9 x π ( x x + 9x - x5 π ( ( ( + 9(- - (5 - π ( π ( π ( 78 - π ( π π Jwnny E UN.! 4. Nili (sin os C. E. - D. Jw: sin( x + - os (x+ + os( x + sin (x+ +.! (sin os (. os sin. -(os π os (sin! - sin Jwnny B UN- 5. Hsil dri E., (,., - (- ( C C. + C (,., 7 (,., 7 + C (,., 6 + C D. + C (,., 7 (,., 7 Jw: misl : u +7 du (6x (x (x du, (,.,. * 6 6 * * Jwnny D. u- 6 + C + C 7 + C (,.,

11 UN 6. Nili dri D. E. C. Jw: (4 +5 x - x + 5x ( - ( - + 5( Jwnny E + -

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10. cos (ax+b)sin(ax+b) dx = 12. sec x dx = tan x + c. 13. sec (ax+b)dx = tan (ax+b)+ c. 14. c sec x dx = - ctg x + c

10. cos (ax+b)sin(ax+b) dx = 12. sec x dx = tan x + c. 13. sec (ax+b)dx = tan (ax+b)+ c. 14. c sec x dx = - ctg x + c BAB XVI. INTEGRAL A. Integrl Tk Tentu. Rumus Integrl Fungsi Aljr. k k n = n +. ( + ) n = ( n + ). = ln + n + + ; n - n+ (+) + ; dn n -. ( f ( ) ± g( ) ) f ( ) ± g ( ) n. os (+)sin(+) = ( n + ) os n + (+)