If [x] denotes the greatest integer less than or equal to x then the value of the integral ${\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} \left[\right[\mathrm{x}]-\sin \mathrm{x}]\mathrm{dx}$ is equal to

$\mathrm{I}={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} \left(\right[\mathrm{x}]+[-\sin \mathrm{x}\left]\right)\mathrm{dx}$           ...(i)

$\begin{array}{c}\mathrm{Using} \mathrm{property}, {\int }_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})\mathrm{dx}\\ \mathrm{I}={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} \left(\right[-\mathrm{x}]+[\sin \mathrm{x}\left]\right)\mathrm{dx} ...\left(\mathrm{ii}\right)\end{array}$

Add Eqs. (i) and (ii),

$\begin{array}{c}2\mathrm{l}={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} \left(\right[\mathrm{x}]+[-\mathrm{x}\left]\right)+\left(\right[\sin \mathrm{x}]+[-\sin \mathrm{x}\left]\right)\mathrm{dx}\\ ={\left.{\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} (-2)\mathrm{dx}=-2\mathrm{x}\right]}_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}[\because [-\mathrm{x}]+[\mathrm{x}]=-1,\text{ if }\mathrm{x}\notin 1]\\ 2\mathrm{I}=(-2)\left(\frac{\mathrm{\pi }}{2}\right)+\left(2\right)\left(-\frac{\mathrm{\pi }}{2}\right)=-2\mathrm{\pi }\end{array}$

$\Rightarrow \mathrm{I}=-\mathrm{\pi }$