∫ 0 20 π sin x + cos x 2 d x is equal to:

Let $I={\int }_0^{20\pi }{\left(\left|\sin x\right|+\left|\cos x\right|\right)}^{2}dx$

We know that the period of $\left|\sin x\right|+\left|\cos x\right|$ is $\frac{\mathrm{\pi }}{2}$

i.e. $I=40{\int }_0^{\frac{\pi }{2}}{\left(\sin x+\cos x\right)}^{2}dx$

$I=40{\int }_0^{\frac{\pi }{2}}\left(1+\sin 2x\right)dx$

$=40{\left(x-\frac{\cos 2x}{2}\right)}_0^{\frac{\mathrm{\pi }}{2}}=40\left(\frac{\mathrm{\pi }}{2}-\frac{\mathrm{cos\pi }}{2}+\frac{\cos 0}{2}\right)$

$I=20\left[\pi +2\right]$