Question

If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.

Options

• 0

• 1

• 2

• 3

Answer 1

If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to 3.

Explanation:

Given: `int_(-a)^a(|x| + |x - 2|)dx` = 22 (a > 2)

|x| = `{{:(x",", x ≥ 0),(-x",", x < 0):}`

|x – 2| = `{{:(x - 2;, x ≥ 2),(-(x - 2);, x < 2):}`

`int_(-a)^af(x)dx = 2int_0^af(x)dx` if `f(-x) = f(x)`

∴ `int_(-a)^a(|x| + |x - 2|)dx = int_(-a)^a|x|dx + int_(-a)^2|x - 2|dx`

= `2int_0^axdx + int_(-a)^2(2 - x)dx + int_2^a(x - 2)dx`

= `2(x^2/2)_0^a + (2x - x^2/2)_(-a)^2 + (x^2/2 - 2x)_2^a`

= `(2a^2)/2 + (2a + a^2/2) + a^2/2 - 2a + 4 - 2 - 2 + 4`

= 2a² + 4

2a² + 4 = 22  ...(Given)

⇒ a² = 9

⇒ a = 3  ...(∵ a > 2)

I = `int_a^(-a)(x + [x])dx`

= `int_3^(-3)(x + [x])dx`

∵ `int_a^bf(x)dx = -int_b^af(x)dx`

∴ I = `-int_(-3)^3xdx - int_(-3)^3[x]dx`

`int_(-a)^af(x)dx` = 0 if `f(-x) = -f(x)`

∴ I' = `int_(-3)^3xdx` = 0

∴ I = `0 - int_(-3)^3[x]dx`

= `[0 - int_(-3)^(-2)(-3)dx + int_(-2)^(-1)(-2)dx + int_(-1)^0(-1)dx + int_0^1 0.dx + int_1^2 1dx + int_2^3 2dx]`

= 0 – [–{3(–2 + 3)2(–1 + 2) + (0 + 1)} + (0 + 1 + 2)}

= –[–3 – 2 – 1 + 3]

= 3