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
\[uv = \int u \frac{\mathrm{d}v}{\mathrm{d}x} \, \mathrm{d}x + \int v \frac{\mathrm{d}u}{\mathrm{d}x} \, \mathrm{d}x\]
Rearranging the terms, we have
\[\int u \frac{\mathrm{d}v}{\mathrm{d}x} \, \mathrm{d}x = uv - \int v \frac{\mathrm{d}u}{\mathrm{d}x} \, \mathrm{d}x\]
Integrating using this formula is called integration by parts.

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

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

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

Integration by parts: motivation from product rule

Integration by parts: motivation from product rule