Integration by parts

This article details the method known as Integration by Parts.

Integration by Parts

Integration by parts is a special 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:

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

This rule is often useful when one function is a power of x and the other function is a trigonometric function or e raised to a 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:

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

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:

<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:
<math>(x^4)(-cos(x)) = -x^4cos(x)</math>

So then the integral becomes:

<math>\int x^4 sin(x)\,dx</math> = <math>\ -x^4cos(x)-(-4x^3sin(x))+(12x^4cos(x))-(-24xsin(x))+(24cos(x))</math>=<math>\ -x^4cos(x)+4x^3sin(x)+12x^2cos(x)+24xsin(x)-24cos(x)+c</math>

See Also

• Integration
• Methods of integration