Evaluate the following integrals:

Let $I=\int x \operatorname{cosec}^{2} x d x$

Using integration by parts,

$I=x \int \operatorname{cosec}^{2} x d x-\int \frac{d}{d x} x \int \operatorname{cosec}^{2} x d x$

We know that, $\int \operatorname{cosec}^{2} x d x=-\cot x$ and $\int \cot x d x=\log |\sin x|$

$=x \times-\cot x-\int-\cot x d x$

$=-x \cot x+\log |\sin x|+c$