Question

${\int }_0^{\infty }\left[{\cot }^{-1}x\right]dx$(where $[\cdot ]$ denotes greatest integer function).

Answer 1

${\int }_0^{\infty }\left[{\cot }^{-1}x\right]dx$

$\left[{\cot }^{-1}x\right]=\{\begin{array}{c}0;\cot 1<x<0\\ 1;\cot 2<x<\cot 1or\cot \frac{\pi }{2}<x<\cot 1\end{array}$

${\int }_0^{\infty }\left[{\cot }^{-1}x\right]dx$

$={\int }_{\cot \frac{\pi }{2}}^{\cot 1}1.dx+{\int }_{\cot 1}^{\cot 0}0.dx$

$={\left[x\right]}_{\cot \frac{\pi }{2}}^{\cot 1}+0$

$=\cot 1-\cot \frac{\pi }{2}$

$=\cot 1-0=\cot 1$