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

Theorem

 * $\displaystyle \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$

Proof
From Fourier Series of $\map \ln {\sin x}$ from $0$ to $\pi$:


 * $\displaystyle \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$

Then, by Parseval's Theorem:

We then have:

So we have:


 * $\displaystyle \frac 4 \pi \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = 2 \paren {\ln 2}^2 + \frac {\pi^2} 6$

multiplying by $\dfrac \pi 4$ we have:


 * $\displaystyle \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$