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

$I={\displaystyle \underset{0}{\overset{\pi }{\int }}\left[\cot \left(\pi -x\right)\right]}dx={\displaystyle \underset{0}{\overset{\pi }{\int }}\left[-\cot x\right]dx}\cdots \cdots \left(2\right)$

(1) + (2)

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

$={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(\left[\cot x\right]+\left(-\cot x\right)\right)dx}$

$2I={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(-1\right)dx}$

$2I={\left(-x\right)}_0{}^{\pi }$

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

$\left[y\right]+\left[-y\right]=-1$

Ex :  $y=2.8$

$2+\left(-3\right)$

$=-1$