What is $\int\log(\sin x)~dx$? I know that the value of the integral of $\cot(x)$ is $\log|\sin x|+C$ .
But what about: 
$$\int\log(\sin x)~dx$$
Is there any easy way to find an antiderivative for this? Thanks.

using $$\ln \sin x =-\ln 2-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n}, \ x \in [0,\pi].$$
$$\int \ln(\sin x)dx=-\ln(2)\int dx-\sum^{\infty}_{n=1}\frac{1}{n}\int (\cos 2nx)dx$$
$$=-\ln(2)\cdot x-\sum^{\infty}_{n=1}\frac{\sin(2nx)}{2n^2}+\mathcal{C}$$
