Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx

Theorem
For $n \in \N_{>0}$:


 * $\ds \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \rd x = -\frac \pi {4 n}$

Proof
First we have:

hence $\ds \lim_{x \mathop \to 0} \map \ln {\sin x} \sin 2 n x = 0$.

Thus: