Explanation

(d)
We know that, $|x| = \left\{ \begin{matrix}
- x,\ \ \ if\ \ \ x < 0\ \\
x,\ \ \ \ if\ \ \ \ \ x \geq 0
\end{matrix} \right.\$
$\therefore\int_{- 2}^{2}{\left( x - |x| \right)dx = \int_{- 2}^{0}{\left\{ \ \right.\ x - ( - x)\} dx + \int_{0}^{2}{(x - x)dx}}}$
$= \int_{- 2}^{0}{2x\ dx + 0}$
$= 2\left\lbrack \frac{x^{2}}{2} \right\rbrack_{- 2}^{0}$
$= 2\left\lbrack 0 - \frac{( - 2)^{2}}{2} \right\rbrack = - 4$