# Difference between revisions of "Integration by parts"

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== Examples == | == Examples == | ||

− | <math>\int xe^x=?</math> | + | <math>\int xe^x\; dx=?</math> |

x has a pretty simple derivative, so let's say <math>u=x</math>. Then <math>dv=e^x dx</math>, <math>du=dx</math>, and <math>v=\int dv=e^x</math>. We have | x has a pretty simple derivative, so let's say <math>u=x</math>. Then <math>dv=e^x dx</math>, <math>du=dx</math>, and <math>v=\int dv=e^x</math>. We have | ||

− | <math>\int xe^x=(x)(e^x)-\int (e^x)(dx)=xe^x-e^x=e^x(x-1)</math>. You can take the derivative to see that it is indeed our desired result. | + | <math>\int xe^x\; dx=(x)(e^x)-\int (e^x)(dx)=xe^x-e^x=e^x(x-1)</math>. You can take the derivative to see that it is indeed our desired result. |

− | Compute <math>\int \tan^{-1}{x}</math>. | + | Compute <math>\int \tan^{-1}{x}\; dx</math>. |

## Revision as of 01:01, 11 July 2006

The purpose of integration by parts is to replace a difficult integral with an easier one. The formula is

## Order

Now, given an integrand, what should be u and what should be dv? Since u will show up as du and dv as v in the integral on the RHS, u should be chosen such that it has an "easy" (or "easier") derivative and dv so that it has a easy antiderivative.

A mnemonic for when to substitute u for what is LIATE:

**L**ogarithmic

**I**nverse trigonometric

**A**lgebraic

**T**rigonometric

**E**xponential

If any two of these types of functions are in the function to be integrated, the type higher on the list should be substituted as u.

## Examples

x has a pretty simple derivative, so let's say . Then , , and . We have

. You can take the derivative to see that it is indeed our desired result.

Compute .