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

Theorem

 * $\ds \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x = \ln 2 - 1$

Proof
It remains to compute:


 * $\ds \lim_{x \mathop \to 0^+} \paren {\map \ln {\sin x} - \map \ln {\tan \frac x 2} }$

We have:

giving:


 * $\ds \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x = \ln 2 - 1$