What methods can be used to solve $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $ I'm seeking methods to solve the following definite integral:
$$ I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $$

Another method I found was to employ integration by parts:
$$\int uv' = uv - \int vu'$$
Let:
\begin{align}
 u&= x & u'&= 1 \\
 v'&= \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} & v' &= \ln|\sin(x)| \\
\end{align}
Thus, 
\begin{align}
\int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)}\:dx &= \bigg[x \ln|\sin(x)| \bigg]_{0}^{\frac{\pi}{2}} - \int_{0}^{\frac{\pi}{2}}\ln|\sin(x)|\:dx \\
 &= - \int_{0}^{\frac{\pi}{2}}\ln|\sin(x)|\:dx
\end{align}
From here, one can source the methods as discussed: here
