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:


Inverse 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.


$\int xe^x\; dx=?$

$x$ 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

$\int xe^x\; dx=(x)(e^x)-\int (e^x)(dx)=xe^x-e^x + C=e^x(x-1) + C$, where $C$ is the constant of integration. You can take the derivative to see that it is indeed our desired result.

See also