${\int }_0^{\pi }[\cot x]dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to:

Let $I={\int }_0^{\pi }[\cot x]dx$.Using the property ${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$:$I={\int }_0^{\pi }[\cot (\pi -x)]dx={\int }_0^{\pi }[-\cot x]dx$.Adding the two:$2I={\int }_0^{\pi }([\cot x]+[-\cot x])dx$.We know $[z]+[-z]=-1$ if $z\notin \mathbb{Z}$ and $0$ if $z\in \mathbb{Z}$.Since $\cot x$ is an integer only at specific points, the integrand is $-1$ almost everywhere.$2I={\int }_0^{\pi }(-1)dx=-\pi ⟹I=-\frac{\pi }{2}$.