Explanation

(c)
Let $I = \int_{0}^{1}{x\left| x - \frac{1}{2} \right|dx}$
$= - \int_{0}^{1/2}{x\left( x - \frac{1}{2} \right)dx + \int_{1/2}^{1}{x\left( x - \frac{1}{2} \right)dx}}$
$= \left\lbrack \frac{x^{2}}{4} - \frac{x^{3}}{3} \right\rbrack_{0}^{1/2} + \left\lbrack \frac{x^{3}}{3} - \frac{x^{2}}{4} \right\rbrack_{1/2}^{1}$
$= \left( \frac{1}{16} - \frac{1}{24} \right) + \left( \frac{1}{3} - \frac{1}{4} - \frac{1}{24} + \frac{1}{16} \right) = \frac{1}{8}$