Integration by Parts

The product rule rearranges to \(\int u\,\mathrm dv=uv-\int v\,\mathrm du\). Choose \(u\) so that differentiating it makes the remaining integral simpler.

A logarithm or inverse circular function often supplies \(u\); use \(\mathrm dv=\mathrm dx\) for \(\int\ln x\,\mathrm dx\) or \(\int\arcsin x\,\mathrm dx\).

For a polynomial times an exponential or trigonometric function, parts may need repeating. For \(\int e^x\sin x\,\mathrm dx\), parts twice returns the original integral; collect it algebraically.

Differentiate the final antiderivative. Preserve brackets and the integration constant \(C\).