Definite Integral from 0 to Pi of x by Logarithm of Sine x

Theorem

 * $\displaystyle \int_0^\pi x \map \ln {\sin x} \rd x = -\frac {\pi^2} 2 \ln 2$

Proof
So:

giving:


 * $\displaystyle \int_0^\pi x \map \ln {\sin x} \rd x = -\frac {\pi^2} 2 \ln 2$