calculus - Integrate $int frac{ln(sin x)}{sin^2 x},mathrm dx.$

I'm having trouble evaluating the integral

$$\int \frac{\ln(\sin x)}{\sin^2 x}\,\mathrm dx.$$

I tried $u$-substitution and integration by parts but they didn't work.

Answer

Here is a start. using integration by parts,

$$ \int u dv = u v - \int v du .$$

Let

$$ u=\ln(\sin(x)) \implies u'=\frac{\cos(x)}{\sin(x)}=\cot(x),\quad v=\int \frac{dx}{\sin^2 x}=-\cot(x). $$

Can you finish it now?