L2 Integration By Parts II - dx 1 2 x 2 ln j 1 x j 1 4 x 2 1 2 x 1 2 ln j 1 x j c ex Z x sec 2(2 x dx 1 2 tan(2 x c ex Z x arctan xdx 1 2 x 2 arctan x 1

L2 Integration By Parts II - dx 1 2 x 2 ln j 1 x j 1 4 x 2...

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Lecture 2: Techniques for Integration(I) Integration by Parts, part II Z udv = uv Z vdu ex. Z ln xdx How to choose u ? ( x ln x x + c ) ex. Evaluate Z 1 0 arctan xdx ( x arctan x + ln(1 = ( p 1 + x 2 )) = = 4 (1 = 2) ln 2) 1
ex. Z e x sin xdx ([ e x (sin x cos x )] = 2 + c ) ex. Z cos x ln(sin x ) dx (sin x ln(sin x )) sin x + c ) 2
Now You Try It (NYTI): ex. Z sin( p x ) dx ( 2 p x cos( p x ) + 2 sin( p x ) + c ) ex. Z x 5 x dx ( x 5 x ln 5 5 x (ln 5) 2 + c ) ex. Z x 5 ln xdx ( x 6 6 ln x 1 6 x 6 6 + c ) ex. Z t 0 e x sin( t x ) dx ( 1 2 ( e t cos t sin t )) 3
ex. Z x ln(1 + x ) dx ( 1 2 x 2 ln j 1 + x j 1 4 x 2 + 1 2 x 1 2 ln j 1 + x j + c ) ex. Z x sec 2 (2 x ) dx ( 1 2 tan(2 x ) + c ) ex. Z x arctan xdx ( 1 2 x 2 arctan x 1 2 x + 1 2 arctan x + c ) Work out NYTI and Webassign integration by parts problems 4

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