As I have found the integral
$$\int_{0}^{\frac{\pi}{2}} \ln (\sin x) d x=-\frac{\pi}{2}\ln 2 ,$$
I started to investigate a similar integral with a different upper limit $\frac{\pi}{4}$ by similar methods with failure. Once I found the Fourier Series of sine on $[0, \pi]$, I can find it easily.
By Fourier Series
$$\ln (\sin x)=-\ln 2-\sum_{n=1}^{\infty} \frac{\cos (2 n x)}{n},$$
we integral from $0$ to $\frac{\pi}{4} $,
\begin{aligned}
\int_{0}^{\frac{\pi}{4}} \ln (\sin x) d x&=-\int_{0}^{\frac{\pi}{4}} \ln 2 d x-\sum_{n=1}^{\infty} \int_{0}^{\frac{\pi}{4}} \frac{\cos (2 n x)}{n} d x \\
&=-\frac{\pi}{4} \ln 2-\sum_{n=1}^{\infty}\left[\frac{\sin (2 n x)}{2 n^{2}}\right]_{0}^{\frac{\pi}{4}} \\
&=-\frac{\pi}{4} \ln 2-\frac{1}{2} \sum_{n=1}^{\infty}\left(\frac{\sin \frac{n \pi}{2}}{n^{2}}\right)\\
&=-\frac{\pi}{4} \ln 2-\frac{1}{2} \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k+1)^{2}} \\
&=-\frac{\pi}{4} \ln 2-\frac{1}{2} G,
\end{aligned}
where $G$ is the Catalan's constant.
Is there any method other than using Fourier Series?
Best Answer
Let $I_0=\int_0^\frac{\pi}{2}\ln(\sin x) dx$. Making the substitution $t=\frac{\pi}{2}-x\,,\,\, I_0=\int_0^\frac{\pi}{2}\ln(\cos x) dx$ $$2I_0=\int_0^\frac{\pi}{2}\ln(\sin x) dx+\int_0^\frac{\pi}{2}\ln(\cos x) dx=\int_0^\frac{\pi}{2}\ln(\cos x\sin x) dx$$ $$=\int_0^\frac{\pi}{2}\ln\frac{1}{2}\,dx\,+\int_0^\frac{\pi}{2}\ln(\sin 2x) dx=-\frac{\pi}{2}\ln 2+I_0\,\,\Rightarrow\,\, I_0=-\frac{\pi}{2}\ln 2$$ Exactly the same approach can be applied to the integrals $I_1=\int_0^\frac{\pi}{4}\ln(\sin x) dx\,$ and $\,I_2=\int_0^\frac{\pi}{4}\ln(\cos x) dx$ $$I_1+I_2=\int_0^\frac{\pi}{4}\ln\frac{1}{2}\,dx\,+\int_0^\frac{\pi}{4}\ln(\sin 2x) dx=-\frac{\pi}{4}\ln 2-\frac{\pi}{4}\ln 2=-\frac{\pi}{2}\ln 2$$ $$I_1-I_2=\int_0^\frac{\pi}{4}\ln(\tan x) dx$$ Making the substitution $x=\operatorname{arctan}t$ $$I_1-I_2=\int_0^1\frac{\ln t}{1+t^2}dt=-G$$
From two equations we get $$I_1=\int_0^\frac{\pi}{4}\ln(\sin x) dx=-\frac{\pi}{4}\ln 2-\frac{G}{2}$$ $$I_2=\int_0^\frac{\pi}{4}\ln(\cos x) dx=-\frac{\pi}{4}\ln 2+\frac{G}{2}$$