Question

Write the value of $\int \frac{\cot x}{\log \left(\sin x\right)}dx$.

Answer 1

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

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

$\Rightarrow \frac{\cos x}{\sin x}dx=dt$

i.e. $\cot xdx=dt$

$\Rightarrow I=\int \frac{dt}{t}$

$=\log \left(t\right)+c$

$=\log \left(\log \right(\sin x\left)\right)+c$