Official solution
\[I = \int\limits_0^{20\pi } {{{\left( {|\sin x| + |\cos x|} \right)}^2}\,dx} \]
\[ = 20\int\limits_0^\pi {\left( {1 + |\sin 2x|} \right)\,dx} \]
\[ = 40\int\limits_0^{{\pi \over 2}} {(1 + \sin 2x)\,dx} \]
\[ = \left. {40\left( {x - {{\cos 2x} \over 2}} \right)} \right|_0^{{\pi \over 2}}\]
\[ = 40\left( {{\pi \over 2} + {1 \over 2} + {1 \over 2}} \right) = 20(\pi + 2)\]