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Integration by parts (view source)
Revision as of 19:13, December 30, 2008
, 19:13, December 30, 2008indented equations for clarity
The proof for Integration by Parts is simple and important.
The proof for Integration by Parts is simple and important.
Given:
Given:
:<math>f(x)g(x)</math>
<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>
It is known from the [[Product rule]] that:
<math>(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)</math>
<br />
<br />
If both sides are integrated:
If both sides are integrated:
:<math>f(x)g(x) = \int f'(x)g(x) + \int f(x)g'(x)</math>
<math>f(x)g(x) = \int f'(x)g(x) + \int f(x)g'(x)</math>
<br />
<br />
By rearranging the terms:
By rearranging the terms:
:<math>\int f(x)g'(x) = f(x)g(x) - \int f'(x)g(x)</math>
<math>\int f(x)g'(x) = f(x)g(x) - \int f'(x)g(x)</math>
==Rapid Repeated Integration==
==Rapid Repeated Integration==
The first term, for example, is:
The first term, for example, is:
:<math>(x^4)(-cos(x)) = -x^4cos(x)</math>
<math>(x^4)(-cos(x)) = -x^4cos(x)</math>
So then the integral becomes:
So then the integral becomes: