Today a student asked me $\int \ln (\sin x) \, dx.$

Calculate the integral
$$\int \ln (\sin x) \, dx.$$

Solutions Collecting From Web of "Today a student asked me $\int \ln (\sin x) \, dx.$"

Consider the following.
\begin{align}
I &= \int \ln(\sin(x)) \ dx
\end{align}
can be evaluated by integration by parts and leads to
\begin{align}
I &= x \ln(\sin(x)) – \int x \ \cot(x) \ dx \\
&= x \ln(\sin(x)) -x \ln(1 – e^{ix}) – \frac{i}{2} \left( x^2 + \operatorname{Li}_2(e^{2ix}) \right)
\end{align}
where $i =\sqrt{-1}$ and $\operatorname{Li}_2(z)$ is the dilogarithm function. It is of note that
\begin{align}
\int_0^{\pi/2} \ln(\sin(x)) \ dx = \frac{\pi}{2} \ln(2).
\end{align}