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

Solution

Maths sir · 4 steps

First use the 22 to find a, then pair x with −x so that [x] + [−x] collapses to −1.

1. Find a. ∫₋ₐᵃ|x| dx = a². For |x-2|, split at x = 2. The graph gives two triangles with areas (a+2)²/2 and (a-2)²/2, which add to a² + 4.

   a² + (a² + 4) = 22 ⇒ a² = 9 ⇒ a = 3

2. The x part. x is an odd function, so ∫₋₃³ x dx = 0.
3. The [x] part. [x] is not odd. Use ∫₋₃³ [x] dx = ∫₀³([x] + [-x])dx. For non-integer x, [x] + [-x] = -1, so this equals -3. Adding unit strips gives the same result: -3-2-1+0+1+2 = -3.
4. So. ∫₋₃³(x + [x])dx = 0 + (-3) = −3.

The common slip: 0. Treating [x] as an odd function and cancelling it over [-3, 3] gives 0, but [-x] ≠ -[x] for non-integers.

The idea to keep. On a symmetric interval, write ∫₋ₐᵃ f = ∫₀ᵃ (f(x) + f(-x))dx. For the greatest integer function, [x] + [-x] = -1 except at integers.

Answer from the NTA final answer key. Solution by PrepWiser.