Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{l}{\int }_0^2\left|\left|x-1\right|-x\right|dx={\int }_0^1\left|1-x-x\right|dx+{\int }_1^2\left|x-1-x\right|dx\end{array}$

$={\int }_0^{\frac{1}{2}}\left(1-2x\right)dx+{\int }_{\frac{1}{2}}^1\left(2x-1\right)dx+{\int }_1^{2}dx$

$={\left(x-x^2\right)}_0^{\frac{1}{2}}+{\left(x^2-x\right)}_{\frac{1}{2}}^1+{\left(x\right)}_1^2=\frac{1}{2}-\frac{1}{4}-\left(0-0\right)+\left(1-1\right)-\left(\frac{1}{4}-\frac{1}{2}\right)+2-1$

$=\frac{1}{4}+\frac{1}{4}+1=\frac{3}{2}=1.5$