Integration by Parts

When the integrand is a product of two functions that is not a substitution pattern — such as $x\,e^{x}$ or $x\log x$ — integration by parts is the technique to use. It comes directly from the product rule for differentiation, read backwards.

The formula

If $u$ and $v$ are both functions of $x$, then

or equivalently, if the product is written as $u\cdot w$ where $w\,dx=dv$,

The idea is to differentiate one factor ($u$) and integrate the other ($dv$), turning a hard integral into $uv$ minus an easier integral.

Choosing which factor is $u$

Choose $u$ to be the factor that becomes simpler when differentiated, and $dv$ to be the factor you can integrate easily. A helpful order of preference for $u$ (functions earlier in the list are chosen first) is:

L – I – A – T – E: Logarithmic, Inverse-trigonometric, Algebraic, Trigonometric, Exponential.

For $x\log x$, the logarithmic factor $\log x$ is chosen as $u$; for $x\,e^{x}$, the algebraic factor $x$ is chosen as $u$ (there is no log or inverse factor, so the algebraic one wins over the exponential).