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Integration by parts (view source)
Revision as of 03:24, December 30, 2008
, 03:24, December 30, 2008added the proof for integration by parts
Note that it may be necessary to repeat the '''integration by parts''' several times, one for each power of ''x''.
Note that it may be necessary to repeat the '''integration by parts''' several times, one for each power of ''x''.
==Proving Integration by Parts==
The proof for Integration by Parts is simple and important.
Given:
<br />
<math>f(x)g(x)</math>
<br />
It is known from the [[Product rule]] that:
<br />
<math>(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)</math>
<br />
If both sides are integrated:
<br />
<math>f(x)g(x) = \int f'(x)g(x) + \int f(x)g'(x)</math>
<br />
By rearranging the terms:
<br />
<math>\int f(x)g'(x) = f(x)g(x) - \int f'(x)g(x)</math>
==Rapid Repeated Integration==
==Rapid Repeated Integration==