Primitive of Sine x by Logarithm of Sine x

Theorem

 * $\displaystyle \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$

Proof
We have:

We also have, by Primitive of Sine Function:


 * $\displaystyle \int \sin x \rd x = -\cos x + C$

So: