Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma

Lemma for Definite Integral from 0 to Half Pi of Logarithm of Sine x

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

Proof
We have:

We can therefore write: