Explanation

(c)
By definition sgn $\left( x - \lbrack x\rbrack \right) = \left\{ \begin{matrix}
\begin{matrix}
\ \\
1,\ \ \ \ \ x > 0
\end{matrix} \\
0,\ \ \ \ x = 0 \\
- 1,\ \ \ x < 0
\end{matrix} \right.\$
$\therefore\int_{- 2}^{10}{sgn\ \left( x - \lbrack x\rbrack \right)dx = - \int_{- 2}^{0}{1\ dx + \int_{0}^{10}{1\ dx}}}$
$= - \left\lceil x \right\rceil_{- 2}^{0} + \lbrack x\rbrack_{0}^{10} = 8$