Explanation

(c)
Let $I = \int_{- 3}^{2}{\left( |x + 1| + |x + 2| + |x - 1 \middle| \ \right)dx}$
Again, let $f(x) = |x + 1| + |x + 2| + |x - 1|$
$= \left\{ \begin{matrix}
\begin{matrix}
\ \\
- (x + 1) - (x + 2) - (x - 1),\ \ - 3 < x \leq - 2 \\
\begin{matrix}
- (x + 1) + x + 2 - (x - 1),\ \ \ \ - 2 < x \leq - 1 \\
1 + x + x + 2 - (x - 1)\ \ \ \ \ - 1 < x \leq 0
\end{matrix}
\end{matrix} \\
1 + x + x + 2 - (x - 1)\ \ \ \ \ 0 \leq x < 1 \\
1 + x + x + 2 + x - 1\ \ \ \ \ 1 \leq x < 2
\end{matrix} \right.\$
$= \left\{ \begin{matrix}
- 3x - 2,\ \ \ \ \ \ - 3 < x \leq - 2 \\
\begin{matrix}
- x + 2,\ \ - 2 < x \leq - 1 \\
x + 4,\ \ - 1 \leq x < 1 \\
3x + 2,\ \ \ \ \ \ 1 \leq x < 2
\end{matrix}
\end{matrix} \right.\$
$\therefore I = \int_{- 3}^{- 2}{( - 3 - 2)dx + \int_{- 2}^{- 1}{( - x + 2)dx}}$
$+ \int_{- 1}^{1}{(x + 4)dx + \int_{1}^{2}{(3x + 2)dx}}$
$= \left\lbrack - \frac{3x^{2}}{2} - 2x \right\rbrack_{- 3}^{- 2} + \left\lbrack - \frac{x^{2}}{2} + 2x \right\rbrack_{- 2}^{- 1}$
$+ \left\lbrack \frac{x^{2}}{2} + 4x \right\rbrack_{- 1}^{1} + \left\lbrack \frac{3x^{2}}{2} + 2x \right\rbrack_{1}^{2}$
$= \left\lbrack - 6 + 4 - \left( - \frac{27}{2} + 6 \right) \right\rbrack + \left\lbrack - \frac{1}{2} - 2 - ( - 2 - 4) \right\rbrack$
$+ \left\lbrack \frac{1}{2} + 4 - \left( \frac{1}{2} - 4 \right) \right\rbrack\left\lbrack 6 + 4 - \left( \frac{3}{2} + 2 \right) \right\rbrack$
$= \frac{11}{2} + \frac{7}{2} + 8 + \frac{13}{2}$
$= \frac{31}{2} + 8 = \frac{47}{2}$
Alternate
Let $I = \int_{- 3}^{2}{\left\{ |x + 1| + |x + 2| + |x - 1| \right\} dx}$
$= \int_{- 3}^{- 1}{|x + 1|dx + \int_{- 1}^{2}{|x + 1|dx + \int_{- 3}^{- 2}{|x + 2|dx}}}$
$+ \int_{- 2}^{2}{|x + 2|dx + \int_{- 3}^{1}{|x - 1|dx}}$
$+ \int_{1}^{2}{|x - 1|dx}$
$= - \int_{- 3}^{- 1}{(x + 1)dx + \int_{- 1}^{2}{(x + 1)dx - \int_{- 3}^{- 2}{(x + 2)dx}}}$
$+ \int_{- 2}^{2}{(x + 2)dx - \int_{- 3}^{1}{(x - 1)dx + \int_{1}^{2}{(x - 1)dx}}}$
$= - \left( \frac{x^{2}}{2} + x \right)_{- 3}^{- 1} + \left( \frac{x^{2}}{2} + x \right)_{- 1}^{2} - \left( \frac{x^{2}}{2} + 2x \right)_{- 3}^{- 2}$
$+ \left( \frac{x^{2}}{2} + 2x \right)_{- 2}^{2} - \left( \frac{x^{2}}{2} - x \right)_{- 3}^{1} - \left( \frac{x^{2}}{2} - x \right)_{1}^{2}$
$= \frac{47}{2}$