Explanation

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