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

$\int u\, dv=uv-\int v\,du$


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:


Iverse trigonometric




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.


$\int xe^x=?$

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

$\int xe^x=(x)(e^x)-\int (e^x)(dx)=xe^x-e^x=e^x(x-1)$. You can take the derivative to see that it is indeed our desired result.

Compute $\int \tan^{-1}{x}$.

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