If ∫ limits-aa(|x|+|x-2|) d x=22,(a2) and [x] denotes the greatest integer ≤ x, then ∫ limitsa - a ( x +[ x ]) dx is equal to .

Question

If ∫ limits-aa(|x|+|x-2|) d x=22,(a>2) and [x] denotes the greatest integer ≤ x, then ∫ limitsa - a ( x +[ x ]) dx is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

∫ limits-a0(-2 x+2) d x+∫ limits02(x+2-x) d x+∫ limits2a(2 x-2) d x=22 x2-.2 x|0 -a+.2 x|0 2+x2-.2 x|0 a=22 a2+2 a+4+a2-2 a-(4-4)=22 2 a2=18 ⇒ a=3 ∫ limits3-3(x+[x]) d x=-(-3-2-1+1+2)=3

Q. If $- a \int a (∣ x ∣ + ∣ x - 2∣) d x = 22, (a > 2)$ and $[x]$ denotes the greatest integer $\leq x$ , then $a \int - a (x + [x]) d x$ is equal to ______.

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

Q. If −a∫a​(∣x∣+∣x−2∣)dx=22,(a>2) and [x] denotes the greatest integer ≤x, then a∫−a​(x+[x])dx is equal to ______.

−a∫0​(−2x+2)dx+0∫2​(x+2−x)dx+2∫a​(2x−2)dx=22 x2−2x∣0−a​+2x∣02​+x2−2x∣0a​=22 a2+2a+4+a2−2a−(4−4)=22 2a2=18⇒a=3 3∫−3​(x+[x])dx=−(−3−2−1+1+2)=3

Q. If $- a \int a (∣ x ∣ + ∣ x - 2∣) d x = 22, (a > 2)$ and $[x]$ denotes the greatest integer $\leq x$ , then $a \int - a (x + [x]) d x$ is equal to ______.

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