Explanation

(d)
Let $I = \int_{- 2}^{2}{|\lbrack x\rbrack|dx}$
$= \int_{- 2}^{- 1}{\left| \lbrack x\rbrack \right|dx + \int_{- 1}^{0}{\left| \lbrack x\rbrack \right|dx + \int_{0}^{1}{\left| \lbrack x\rbrack \right|dx + \int_{1}^{2}{|\lbrack x\rbrack|dx}}}}$
$= \int_{- 2}^{- 1}{2dx + \int_{- 1\ }^{0}{1dx + \int_{0}^{1}{0dx + \int_{1}^{2}{1dx}}}}$
$= 2\lbrack x\rbrack_{- 2}^{- 1} + \lbrack x\rbrack_{- 1}^{0} + 0 + \lbrack x\rbrack_{1}^{2} = 4$