${\int }_0^{20\mathrm{\pi }} \left(\right|\sin \mathrm{x}|+|\cos \mathrm{x}|{)}^2\mathrm{dx}$ is equal to :-

$\begin{array}{l}\mathrm{I}={\int }_0^{20\mathrm{\pi }} \left(\right|\sin \mathrm{x}|+|\cos \mathrm{x}|{)}^2\mathrm{dx};(\text{ Jack property) }\\ period of |\sin \mathrm{x}|+|\cos \mathrm{x}|=\frac{\pi }{2}\\ \mathrm{I}=40{\int }_0^{\mathrm{\pi }/2} (\sin \mathrm{x}+\cos \mathrm{x}{)}^2\mathrm{dx}\\ \mathrm{I}=40{\int }_0^{\mathrm{\pi }/2} (1+\sin 2\mathrm{x})\mathrm{dx}\\ \mathrm{I}=40{\left(x-\frac{\cos 2x}{2}\right)}_0^{\frac{\mathrm{\pi }}{2}}\\ \mathrm{I}=20(\pi +2)\end{array}$