# Difference between revisions of "Integration by parts"

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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\; 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 | + | <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. |

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== See also == | == See also == |

## 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.