$I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\sin \left(x-\left[x\right]\right)dx$, where $\left[·\right]$ denotes G.I.F.

$\Rightarrow I={\int }_{-\frac{\mathrm{\pi }}{2}}^{-1}\sin \left(2+x\right)dx+{\int }_{-1}^0\sin \left(1+x\right)dx+{\int }_0^1\sin xdx+{\int }_1^{\frac{\mathrm{\pi }}{2}}\sin \left(x-1\right)dx$

$\Rightarrow I=-{\left[\cos \left(x+2\right)\right]}_{-\frac{\mathrm{\pi }}{2}}^{-1}-{\left[\cos \left(x+1\right)\right]}_{-1}^0-{\left[\cos x\right]}_0^1-{\left[\cos \left(x-1\right)\right]}_1^{\frac{\mathrm{\pi }}{2}}$

$\Rightarrow I=-\left[\cos 1-\cos \left(-\frac{\mathrm{\pi }}{2}+2\right)\right]-\left[\cos 1-\cos 0\right]-\left[\cos 1-\cos 0\right]-\left[\cos \left(\frac{\mathrm{\pi }}{2}-1\right)-\cos 0\right]$

$\Rightarrow I=-\cos 1+\sin 2-\cos 1+1-\cos 1+1-\sin 1+1$

$\Rightarrow I=3\left(1-\cos 1\right)+\sin 2-\sin 1$