7.1 Integration by Parts

7.1 Integration by Parts - 7.1 Integration by Parts If u...

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7.1 Integration by Parts If u and v are functions of x and have continuous derivatives, then ∫ ∫ - = du v uv dv u Note: This is the formula for Integration by Parts
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Guidelines for Integration by Parts Try letting 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 than u . Then dv will be the remaining factor(s) of the integrand.
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Example 1 Evaluate the integral: - dx x ) ( sin 1 dx x du x u 2 1 1 1 ) ( sin let - = = - Solution: x v dx dv = = ( 29 C x x x C x x x dx x x x x dx x x x x dx x + - + = + - + = - - = - - = - - - - - - 2 1 2 / 1 2 1 2 / 1 2 1 2 1 1 1 ) ( sin 2 1 2 1 ) ( sin ) 1 ( ) ( sin 1 ) ( sin ) ( sin
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Example 2- Repeated Application of Integration by Parts Evaluate the integral: - θ d e ) 2 cos( d du u ) 2 sin( 2 ) 2 cos( let - = = - - - = = e v d e dv - - - - - = θ d e e d e ) 2 sin( 2 ) 2 cos( ) 2 cos( d dU ) 2 cos( 4 ) 2sin(2 let U = = - - - = = e V d e dV [ ] 5 where ) 2 sin( 5 2 ) 2 cos( 5 1 ) 2 cos( ) 2 sin( 2 )
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7.1 Integration by Parts - 7.1 Integration by Parts If u...

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