Question:
If $\int_{-\mathrm{a}}^{\mathrm{a}}(|\mathrm{x}|+|\mathrm{x}-2|) \mathrm{dx}=22,(\mathrm{a}>2)$ and $[\mathrm{x}]$ denotes the greatest integer $\leq \mathrm{x}$, then $\int_{\mathrm{a}}^{-\mathrm{a}}(\mathrm{x}+[\mathrm{x}]) \mathrm{dx}$ is equal to
Solution:
$\int_{-a}^{0}(-2 x+2) d x+\int_{0}^{2}(x+2-x) d x+\int_{2}^{a}(2 x-2) d x=22$
$x^{2}-\left.2 x\right|_{0} ^{-a}+\left.2 x\right|_{0} ^{2}+x^{2}-\left.2 x\right|_{2} ^{a}=22$
$a^{2}+2 a+4+a^{2}-2 a-(4-4)=22$
$2 a^{2}=18 \Rightarrow a=3$
$\int_{3}^{-3}(x+[x]) d x=-\left(\int_{-3}^{3}(x+[x]) d x\right)=-\left(\int_{-3}^{3}[x] d x\right)$
$=-(-3-2-1+0+1+2)=3$
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