Question

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

Answer 1

$\int\limits_{-a}^{0}(-2 x+2) d x+\int\limits_{0}^{2}(x+2-x) d x+\int\limits_{2}^{a}(2 x-2) d x=22$
$x^{2}-\left.2 x\right|_{0} ^{-a}+\left.2 x\right|_{0} ^{2}+x^{2}-\left.2 x\right|_{0} ^{a}=22$
$a^{2}+2 a+4+a^{2}-2 a-(4-4)=22$
$2 a^{2}=18 \Rightarrow a=3$
$\int\limits_{3}^{-3}(x+[x]) d x=-(-3-2-1+1+2)=3$