Calculus2.5

# Calculus2.5 - Section 7.2 Integration by Parts d(u v u dv v...

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Section 7.2: Integration by Parts

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u u v du d v v ( ) u u v dv d du v v dv u v u du Integrating we get This is the integral version of the product rule!
sin x x dx u x sin dv x dx s ( ) co x x cos sin x x x C u u v du d v v cos dx x du dx cos v x

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Idea Let u be part that gets better when we take derivative For example: Let dv be part that we can integrate For example : , ,ln n x x x sin ,cos , x x x e
2 5 x x e dx 2 2 u x du x dx 5 5 1 5 x x dv e dx v e 2 5 1 5 x e x u u v du d v v 5 2 5 x e x dx

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2 5 5 1 2 5 5 x x x e xe dx u x du dx 5 5 1 5 x x dv e dx v e 5 5 5 2 1 2 1 1 5 5 5 5 x x x e x C dx e x e 5 5 5 2 1 2 1 1 5 5 5 5 2 x x x x x e C e e
3 sin( ) x x dx There is a simple way to keep track 3 2 3 6 6 0 x x x sin cos sin cos sin x x x x x + - + - 3 c ( ) os x x

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