If the value of the integral
∫₀⁵ (x + [x])/e^(x - [x])dx = α e⁻¹ + β , where α, β ∈ R, 5α + 6β = 0, and [x] denotes the greatest integer less than or equal to x; then the value of (α + β)² is equal to :

Solution

Maths sir · 5 steps

Split at every integer: on each piece [x] is fixed, so it becomes one small integral repeated five times.

1. Split at integers. On [n, n+1) we have [x] = n. Put x = n + t with t from 0 to 1, so x - [x] = t and the integrand becomes (2n + t)e⁻ᵗ.

   ∫₀⁵ (x+[x])/e^(x-[x]) dx = Σₙ₌₀⁴ ∫₀¹ (2n+t) e⁻ᵗ dt

2. Two basic integrals. ∫₀¹ e⁻ᵗ dt = 1 - e⁻¹ and, by parts, ∫₀¹ t e⁻ᵗ dt = 1 - 2e⁻¹. Since 2(0+1+2+3+4) = 20, the total is 20(1 - e⁻¹) + 5(1 - 2e⁻¹).
3. Find alpha. The coefficient of e⁻¹ is -20 - 10, so α = -30.
4. Use the condition. The given relation 5α + 6β = 0 gives β = -5α/6 = 25. Note that adding the constant terms directly gives 30, which does not satisfy this relation. The question's data is inconsistent, and the key takes β from the given condition.
5. So. α + β = -30 + 25 = -5, so (α + β)² = 25, option (b).

The idea to keep. When [x] or x - [x] appears, split the range at integers. On [n, n+1) put x = n + t, and every piece turns into the same integral in t from 0 to 1.

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