The value of $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{\left[x\right]+\left[\sin x\right]+4},$ where $\left[t\right]$ denotes the greatest integer less than or equal to t, is 

$\begin{array}{rcl}\mathrm{I}& =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{\mathrm{dx}}{\left[\mathrm{x}\right]+[\sin \mathrm{x}]+4}\\ & =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{-1 } \frac{\mathrm{dx}}{-2-1+4}+{\int }_{-1}^0 \frac{\mathrm{dx}}{-1-1+4}+{\int }_0^1 \frac{\mathrm{dx}}{0+0+4}+{\int }_1^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2 $}\right.}\frac{\mathrm{dx}}{1+0+4}\\ & =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{-1} \mathrm{dx}+{\int }_{-1}^0 \frac{\mathrm{dx}}{2}+{\int }_0^1 \frac{\mathrm{dx}}{4}+{\int }_1^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{\mathrm{dx}}{5}\\ & =& \left(-1+\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)+\frac{1}{2}(0+1)+\frac{1}{4}+\frac{1}{5}\left(\frac{\mathrm{\pi }}{2}-1\right)\\ & =& \frac{3}{20}(4\mathrm{\pi }-3)\end{array}$