$I=\int\limits_{0}^{1}{\left( \left| 2x-1 \right|-\left| x \right| \right)dx}$    

$\Rightarrow I=\int\limits_{0}^{\frac{1}{2}}{\left( -2x+1-x \right)dx}+\int\limits_{\frac{1}{2}}^{1}{\left( 2x-1-x \right)dx}$

$=\left. \left( -\frac{3{{x}^{2}}}{2}+x \right) \right|_{0}^{\frac{1}{2}}+\left. \left( \frac{{{x}^{2}}}{2}-x \right) \right|_{\frac{1}{2}}^{1}$$=\frac{-3}{8}+\frac{1}{2}+\frac{1}{2}-1-\frac{1}{8}+\frac{1}{2}=0$