∫limits_0^π[cot x] d x, [.] denotes the greatest integer function, is…

Let $I=\int\limits_0^\pi[\cot x] d x$          .....(i)

$\Rightarrow I=\int\limits_0^\pi[\cot (\pi-x)] d x$          [Using : $\int\limits_0^a f(x) d x=\int\limits_0^a f(a-x) d x$]

$\Rightarrow I=\int\limits_0^\pi[-\cot x] d x$              ....(ii)

Adding (i) and (ii), we get

$2 I=\int\limits_0^\pi\{[\cot x]+[-\cot x]\} d x$

$\Rightarrow 2 I=\int\limits_0^\pi-1 d x$               [∵ $[x]+[-x]=-1$, if $x \notin Z$]

$\Rightarrow 2 I=-\pi \Rightarrow I=-\frac{\pi}{2}$