Introduction to Integration by Parts

We begin with the Product Rule for differentiation:
\({\frac{\mathrm{d}}{\mathrm{d}x}(uv) = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}}\)

Rearranging the equation gives:
\({u\frac{\mathrm{d}v}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}x}(uv) - v\frac{\mathrm{d}u}{\mathrm{d}x}}\)

Now, integrate both sides with respect to \(x\):
\({\displaystyle\displaystyle \displaystyle\int u\frac{\mathrm{d}v}{\mathrm{d}x},dx = \displaystyle\displaystyle \displaystyle\int \frac{\mathrm{d}}{\mathrm{d}x}(uv),dx - \displaystyle\displaystyle \displaystyle\int v\frac{\mathrm{d}u}{\mathrm{d}x},dx}\)

The integral of the derivative \({\frac{\mathrm{d}}{\mathrm{d}x}(uv)}\) simplifies to \({uv}\), leading to the Integration by Parts formula:
\({\boxed{\displaystyle\displaystyle \displaystyle\int u,dv = uv - \displaystyle\displaystyle \displaystyle\int v,du}}\)

This formula allows us to transform an integral involving a product of two functions into another, often simpler, form.

Key Idea

Not every expression can be directly integrated, but almost any function we’ve learned can be differentiated.
Hence, when applying integration by parts, it is often strategic to differentiate the more complicated function (set it as \(u\)) and integrate the simpler one (set it as \(dv\)).

Let’s find:
\({\displaystyle\displaystyle \displaystyle\int x \ln x \mathrm{ d}x}\)