$\underset{0}{\overset{\pi }{\int }}\left[\cot x\right]dx,$ where $\left[.\right]$ denotes the greatest integer function, is equal to 

$I=\underset{0}{\overset{\pi }{\int }}\left[\cot x\right]dx$

$$\begin{gathered} =\underset{0}{\overset{\pi }{\int }}\left[\cot \left(\pi -x\right)\right]dx \\ I=\underset{0}{\overset{\pi }{\int }}\left[cot\left(\pi -x\right)\right]dx \\ I=\underset{0}{\overset{\pi }{\int }}\left[-cotx\right]dx \\ I+I=\underset{0}{\overset{\pi }{\int }}\left(\left[cotx\right]+\left[-cotx\right]\right)dx \\ 2I=\underset{0}{\overset{\pi }{\int }}\left(-1\right)dx \\ \left(\left[x\right]+\left[-x\right]=-1 \\ if x\notin Z \\ and \left[x\right]+\left[-x\right]=0 \\ if x\in z\right) \\ I=\frac{1}{2}{\left(-x\right)}_0^{\pi } \\ =\frac{1}{2}\left(-\pi \right) \end{gathered}$$