If ∫₀¹ xˣ dx = ∫₀¹ (1 − x)ˣ dx, then the value of α is:

If ∫₀¹ xˣ dx = ∫₀¹ (1 − x)ˣ dx, then the value of α is:

• A. 0
• B. 1
• C. 2
• D. any real number

Answer: D) any real number

Explanation: Let I = ∫₀¹ xˣ dx. Put x = 1 − t, dx = −dt. Limits: 0→1, 1→0. I = ∫₁⁰ (1−t)ˣ (−dt) = ∫₀¹ (1−t)ˣ dt. So equality holds for any α. The variable of integration is dummy; the integral of xˣ from 0 to 1 equals integral of (1−x)ˣ from 0 to 1 by substitution, regardless of α.