$\begin{aligned} & \int_0^\pi\left|\sin x-\frac{2 x}{\pi}\right| \mathrm{d} x \\ & =\int_0^{\frac{\pi}{2}}\left(\sin x-\frac{2 x}{\pi}\right) \mathrm{d} x+\int_{\frac{\pi}{2}}^\pi\left(\frac{2 x}{\pi}-\sin x\right) \mathrm{d} x \\ & =[-\cos x]_0^{\frac{\pi}{2}}-\left[\frac{x^2}{\pi}\right]_0^{\frac{\pi}{2}}+\left[\frac{x^2}{\pi}\right]_{\frac{\pi}{2}}^\pi+[\cos x]_{\frac{\pi}{2}}^\pi \\ & =(0+1)-\left(\frac{\pi}{4}-0\right)+\left(\pi-\frac{\pi}{4}\right)+(-1-0) \\ & =\frac{\pi}{2}\end{aligned}$