$\begin{aligned} & \int_0^{2 \pi}(\sin x+|\sin x|) d x= \\ & =\int_0^\pi(\sin x+\sin x) d x+\int_0^\pi(\sin x-\sin x) d x \\ & =\int_0^\pi 2 \sin x d x+0 \\ & =[-2 \cos x]_0^\pi=4\end{aligned}$