# Difference between revisions of "Integration by parts"

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− | The purpose of integration by parts is to replace a difficult integral with an easier one. The formula is | + | The purpose of integration by parts is to replace a difficult [[integral]] with an easier one. The formula is |

<math>\int u\, dv=uv-\int v\,du</math> | <math>\int u\, dv=uv-\int v\,du</math> | ||

− | |||

− | |||

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

− | A mnemonic for when to substitute u for what is LIATE: | + | A mnemonic for when to substitute <math>u</math> for what is LIATE: |

'''L'''ogarithmic | '''L'''ogarithmic | ||

− | '''I''' | + | '''I'''nverse trigonometric |

'''A'''lgebraic | '''A'''lgebraic | ||

Line 23: | Line 21: | ||

== 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 + C=e^x(x-1) + C</math>, where <math>C</math> is the [[constant of integration]]. You can take the derivative to see that it is indeed our desired result. |

+ | |||

+ | == See also == | ||

+ | * [[Calculus]] | ||

− | + | [[Category:Advanced Mathematics Topics]] |

## Latest revision as of 11:39, 18 November 2010

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

A mnemonic for when to substitute 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

, where is the constant of integration. You can take the derivative to see that it is indeed our desired result.