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'''Integration by parts''' is a special [[Techniques of integration|technique]] to facilitate the integration of the product of two functions that otherwise lack an obvious integral.  This technique can be proven with the [[product rule]].

'''Integration by parts''' is a special [[Techniques of integration|technique]] to facilitate the integration of the product of two functions that otherwise lack an obvious integral.  This technique can be proven with the [[product rule]].

The rule for '''integration by parts''' is stated as follows:

The rule for '''integration by parts''' is stated as follows:<ref>http://www.math.hmc.edu/calculus/tutorials/int_by_parts/</ref>

:<big><math>\int f(x) g'(x)\,dx = f(x) g(x) - \int f'(x) g(x)\,dx,</math></big>

:<big><math>\int f(x) g'(x)\,dx = f(x) g(x) - \int f'(x) g(x)\,dx,</math></big>

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''.

===Rapid Repeated Integration===

==Rapid Repeated Integration==

Rapid Repeated Integration is a shortcut method for reduction problems that require Integration by Parts. It is especially useful when one function's derivative reduces to zero.

'''Rapid Repeated Integration''' is a shortcut method for reduction problems that require Integration by Parts. It is especially useful when one function's derivative reduces to zero.

For example, integrating:

For example, integrating:

:<big><math>\int x^4 sin(x)\,dx</math></big>

:<big><math>\int x^4 sin(x)\,dx</math></big>

*[[Integration]]

*[[Integration]]

*[[Methods of integration]]

*[[Methods of integration]]

[[category:mathematics]]

[[category:mathematics]]

[[category:calculus]]

[[category:calculus]]