Integration by Parts

# Integration by Parts - The Method of Integration by Parts...

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The Method of Integration by Parts

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Main Idea If u & v are differentiable functions of x, then By integrating with respect to x, we get : ' )' ( ' ' ' )' ( vu uv uv vu uv uv - = + = - = - = vdu uv udv dx vu dx uv dx uv ' )' ( '
When to use this method? When the integrand is a product of the form udv, such that we do not know how to find the integral ∫udv, but can find v = ∫dv and the integral ∫vdu.

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Examples I When, we have an integrand, similar to one of the following: ( where b and c are any real numbers) 1. x n cos(cx) or x n sin(cx) ; where n is a natural number 2. x n e cx or x n a cx ; where n is a natural number and b is a base for an exponential function ( b is positive and not equal to 1) 3. x lnx or x c lnx b
Example 1 = dx x x I cos

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c x x x dx x x x vdu uv I x v dx du dx x dv x u Let dx x x I + + = - = - = = = = = = cos sin sin sin sin , .. cos , cos
Example 2 = dx x x I sin 2

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c x x x x x c x x x x x dx x x x x vdu uv I x v xdx du dx x dv x u Let dx x x I + + + - = + + + - = + - = - = - = = = = = cos 2 sin 2 cos ] cos sin [ 2 cos cos 2 cos cos , 2 sin , sin 2 2 2 2 2
Example 3 = dx x x I cos 3

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c x x x x x x x dx x x x x vdu uv I x v dx x du dx x dv x u Let dx x x I + + + - + - = + - = - = = = = = = ] cos 2 sin 2 cos [ 3 cos sin 3 sin sin , 3 cos , cos 2 2 2 2 2 3 3
Example 5 = dx xe I x

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c e xe dx e xe vdu uv I e v and dx du dx e dv and x u Let dx xe I x x x x x x x + - = - = - = = = = = =
Example 5 = dx e x I x 2

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c e xe e x c e xe e x dx xe e x vdu uv I e v and xdx du dx e dv and x u Let dx e x I x x x x x x x x x x x + + - = + - - = - = - = = = = = = 2 2 ] [ 2 2 2 2 2 2 2 2
Example 6 = dx e x I x 3

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c e xe e
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Integration by Parts - The Method of Integration by Parts...

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