Explanation

(c)
Let $I = \int_{- 2}^{4}{|x + 1|dx}$
$= \int_{- 2}^{- 1}{- (x + 1)dx + \int_{- 1}^{4}{(1 + x)dx}}$
$= - \left\lbrack \frac{x^{2}}{2} + x \right\rbrack_{- 2}^{- 1} + \left\lbrack x + \frac{x^{2}}{2} \right\rbrack_{- 1}^{4}$
$= - \left\lbrack \frac{1}{2} - 1 - (2 - 2) \right\rbrack + \left\lbrack 4 + 8—\left( 1 + \frac{1}{2} \right) \right\rbrack$
$= \frac{1}{2} + \left( \frac{25}{2} \right) = 13$