# How do you integrate int x^2 cot2 x dx  using integration by parts?

Mar 18, 2016

As a first step choose $u = {x}^{2}$ and $\mathrm{dv} = \cot x \mathrm{dx}$

#### Explanation:

This gets us $\mathrm{du} = 2 x \mathrm{dx}$ and $v = \frac{1}{2} \ln \left(\sin x\right)$,
so the integral is equal to

$\frac{1}{2} {x}^{2} \ln \left(\sin x\right) - \int x \ln \left(\sin x\right) \mathrm{dx}$

Writing the integral now will involve the polylogarithmic function. You can read more about that here.