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If \int_{-a}^{a}\left ( \left | x \right |+\left | x -1\right | \right )dx=22, ( a > 2 ) and [x] denotes the greatest integer \leq x, then \int_{a}^{-a}\left ( x+\left [ x \right ] \right )dx is equal to:
Option: 1 1
Option: 2 2
Option: 3 3
Option: 4 4

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\\\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=3+2+1+0-1-2=3

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Suraj Bhandari

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