Integration by parts: motivation from product rule

Let \(u\) and \(v\) be two functions of \(x\).

Then the derivative of \(uv\) with respect to \(x\) gives

\[\frac{\mathrm{d}}{\mathrm{d}x}uv = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}.\]

Now integrating both sides with respect to \(x\) gives

\[uv = \displaystyle \displaystyle\int u \frac{\mathrm{d}v}{\mathrm{d}x} \, \mathrm{d}x + \displaystyle \displaystyle\int v \frac{\mathrm{d}u}{\mathrm{d}x} \, \mathrm{d}x\]

Rearranging the terms, we have

\[\displaystyle \displaystyle\int u \frac{\mathrm{d}v}{\mathrm{d}x} \, \mathrm{d}x = uv - \displaystyle \displaystyle\int v \frac{\mathrm{d}u}{\mathrm{d}x} \, \mathrm{d}x\]

Integrating using this formula is called integration by parts.