\int\limits_{-5}^{2}\mathrm{sign}(\sin x)(x-\mathrm{sign}\,x)\,dx=\\
\int\limits_{-5}^{-\pi}1\cdot(x+1)\,dx+
\int\limits_{-\pi}^{0}(-1)\cdot(x+1)\,dx+
\int\limits_{0}^{2}1\cdot(x-1)\,dx=\\
=\pi^2-2\pi-\frac{15}{2}