$\underset{-a}{\overset{a}{\int }}\left(\left|x\right|+\left|x-2\right|\right)dx=]22,\left(a>2\right)$ and [x]denotes the greatest integer , then $\underset{a}{\overset{-a}{\int }}\left(x+\left[x\right]\right)dx$ is equal to 

$\begin{array}{c}\underset{-a}{\overset{0}{\int }}\left(-2x+2\right)dx+\underset{0}{\overset{2}{\int }}\left(x+2-x\right)dx+\underset{2}{\overset{a}{\int }}\left(2x-2\right)dx=22\\ {\left(-x^2+2x\right)}_{-a}^0+{\left[2x\right]}_0^2+{\left[x^2-2x\right]}_2^a=22\\ a^2+2a+4+a^2-2a-\left(4-4\right)=22\\ 2a^2=18\\ a=3\\ \underset{-3}{\overset{3}{\int }}\left(x+\left[x\right]\right)dx=-3-2-1+1+2 (x is an odd function)\\ =-3\end{array}$