Question

Evaluate $\int \frac{\log \left(\sin x\right)}{\tan x}dx$

Answer 1

Let $I=\int \frac{\log \left(\sin x\right)}{\tan x}dx$

Put $\log \left(\sin x\right)=t$

$\frac{1}{\sin x}\cos xdx=dt$

$\cot xdx=dt$

$\frac{1}{\tan x}dx=dt$

Therefore, $I=\int tdt$

$=\frac{t^2}{2}+C=\frac{(\log \sin x{)}^2}{2}+C$