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Integration by parts (view source)
Revision as of 03:00, December 30, 2008
, 03:00, December 30, 2008moved from the methods of integration page
'''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 utilizes the insight of the [[product rule]].
This article details the method known as '''Integration by Parts'''.
==Integration by Parts==
'''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:
:<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 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:
:<big><math>\int x^4 sin(x)\,dx</math></big>
We start off by making a table of <math>x^4</math> and <math>sin(x)</math>. On the first column, the derivatives of <math>x^4</math> are taken until they reach zero. In the second column, sin(x) is integrated once down each row:
{| class="wikitable"
|-
! <math>f(x)</math>
! <math>g(x)</math>
|-
| <math>x^4</math>
| <math>sin(x)</math>
|-
| <math>4x^3</math>
| <math>-cos(x)</math>
|-
| <math>12x^2</math>
| <math>-sin(x)</math>
|-
| <math>24x</math>
| <math>cos(x)</math>
|-
| <math>24</math>
| <math>sin(x)</math>
|-
| <math>0</math>
| <math>-cos(x)</math>
|}
Terms are multiplied diagonally from the left to the right and we add the product to the product of the next product, alternating signs with each step.
The first term, for example, is:
<br />
<math>(x^4)(-cos(x)) = -x^4cos(x)</math>
So then the integral becomes:
:<big><math>\int x^4 sin(x)\,dx</math></big> = <big><math>\ -x^4cos(x)-(-4x^3sin(x))+(12x^4cos(x))-(-24xsin(x))+(24cos(x))</math></big>=<big><math>\ -x^4cos(x)+4x^3sin(x)+12x^2cos(x)+24xsin(x)-24cos(x)+c</math></big>
== See Also ==
*[[Integration]]
*[[Methods of integration]]
[[category:mathematics]]
[[category:mathematics]]
[[category:calculus]]
[[category:calculus]]