If $$\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22$$, (a > 2) and [x] denotes the greatest integer $\le$ x, then$\int\limits_{ - a}^a {\left( {x + \left[ x \right]} \right)} dx$ is equal to _________.

$$\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22$$ <br><br>$\Rightarrow$ $$\int\limits_{ - a}^0 {( - 2x + 2)dx} + \int\limits_0^2 {(x + 2 - x)dx} + \int\limits_2^a {(2x - 2)dx} = 22$$<br><br>$\Rightarrow$ ${x^2} - 2x|_0^{ - a} + 2x|_0^2 + {x^2} - 2x|_2^a = 22$<br><br>$\Rightarrow$ ${a^2} + 2a + 4 + {a^2} - 2a - (4 - 4) = 22$<br><br>$\Rightarrow$ $2{a^2} = 18 \Rightarrow a = 3$<br><br>$$\int\limits_3^{ - 3} {(x + [x])dx} = - \left( {\int\limits_{ - 3}^3 {(x + [x])dx} } \right) = - \left( {\int\limits_{ - 3}^3 {[x]dx} } \right)$$ <br><br>= -($$\int\limits_{ - 3}^{ - 2} {\left[ x \right]dx} + \int\limits_{ - 2}^{ - 1} {\left[ x \right]dx} + \int\limits_{ - 1}^0 {\left[ x \right]dx} $$ <br><br>+ $$\int\limits_0^1 {\left[ x \right]dx} + \int\limits_1^2 {\left[ x \right]dx} + \int\limits_2^3 {\left[ x \right]dx} $$) <br><br>$= - ( - 3 - 2 - 1 + 0 + 1 + 2) = 3$