If ∫ − a a ( | x | + | x − 2 | ) d x = 2 2 , ( a > 2 ) a n d [ x ] denotes the greatest integer ≤ x , then ∫ a − a ( x + [ x ] ) d x is equal to__________.

∫ − a a ( | x | + | x − 2 | ) d x = 2 2 , a > 2 ∫ − a 0 ( − 2 x + 2 ) d x + ∫ 0 2 ( x − x + 2 ) d x + ∫ 2 a ( 2 x − 2 ) d x = 2 2 ⇒ 2 a 2 + 2 = 2 0 ⇒ a 2 = 9 ⇒ a = 3 ∴ ∫ 3 − 3 ( x + [ x ] ) d x = − ∫ − 3 3 ( 2 x − { x } ) d x = − ∫ − 3 3 2 x d x + 6 ∫ 0 1 x d x = 6 . x 2 2 | 0 1 = 3

If $\int_{- a}^{a} (| x | + | x - 2 |) d x = 22, (a > 2) a n d [x]$ denotes the greatest integer $\leq x,$ then $\int_{a}^{- a} (x + [x]) d x$ is equal to__________.

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Vishal Baghel | Contributor-Level 10

4 months ago

$\int_{- a}^{a} (| x | + | x - 2 |) d x = 22, a > 2$

$\int_{- a}^{0} (- 2 x + 2) d x + \int_{0}^{2} (x - x + 2) d x + \int_{2}^{a} (2 x - 2) d x = 22$

$\Rightarrow 2 a^{2} + 2 = 20 \Rightarrow a^{2} = 9 \Rightarrow a = 3$

$∴ \int_{3}^{- 3} (x + [x]) d x = - \int_{- 3}^{3} (2 x - {x}) d x = - \int_{- 3}^{3} 2 x d x + 6 {\int_{0}^{1} x d x = 6 . \frac{x^{2}}{2} |}_{0}^{1} = 3$

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