${\displaystyle {\int }_0^1\left|\text{sin}2\pi x\right|dx=}$

${\displaystyle \underset{0}{\overset{1}{\int }}\left|Sin2\pi x\right|dx}={\displaystyle \underset{0}{\overset{1/2}{\int }}\left|Sin2\pi x\right|dx}+{\displaystyle \underset{1/2}{\overset{1}{\int }}\left|Sin2\pi x\right|}dx$

                          $={\displaystyle \underset{0}{\overset{1/2}{\int }}Sin2\pi xdx}+{\displaystyle \underset{1/2}{\overset{1}{\int }}-Sin2\pi xdx}$

                          $={\left(\frac{-Cos2\pi x}{2\pi }\right)}_0^{\frac{1}{2}}+{\left(\frac{Cos2\pi x}{2\pi }\right)}_{\frac{1}{2}}^1$

                          $=\frac{4}{2\pi }=\frac{2}{\pi }$