∫ _ 0 ^ 20π (| x| + | x|) ^ 2 dx is equal to:

Let $I = {∫}_{0}^{20\pi}{\left(\left|\sin x\right| + \left|\cos x\right|\right)}^{2}dx$

We know that the period of \left|\sin x\right| + \left|\cos x\right|$is$\dfrac{\pi}{2}$</p><p>i.e.$I = 40{∫}_{0}^{\dfrac{\pi}{2}}{\left(\sin x + \cos x\right)}^{2}dx$</p><p>$I = 40{∫}_{0}^{\dfrac{\pi}{2}}\left(1 + \sin 2x\right)dx$</p><p>$= 40{\left(x - \dfrac{\cos 2x}{2}\right)}_{0}^{\dfrac{\pi}{2}} = 40\left(\dfrac{\pi}{2} - \dfrac{cos \pi}{2} + \dfrac{\cos 0}{2}\right)$</p><p>$I = 20 \left[\pi + 2\right]