The integral ${\int }_0^{2c} |x-c|dx$ equal :

By definition:- 

$|x-c|=\left\{\begin{array}{c}c-x\text{ if }0\le x\le c\\ x-c\text{ if c}\le x\le 2c\end{array}\right.$

So if $I={\int }_0^{2c} |x-c|dx$

Then $I={\int }_0^c (c-x)dx+{\int }_c^{2c} (x-c)dx$

$\begin{array}{l}=cx-{\left.\frac{x^2}{2}\right|}_0^c+{\left|\frac{x^2}{2}-cx\right|}_c^{2c}\\ =c^2-\frac{c^2}{2}+2c^2-2c^2-\frac{c^2}{2}+c^2=c^2\end{array}$