If ∫₀ᵃ 1/(1 + 4x²) dx = π/8, then a equals:

If ∫₀ᵃ 1/(1 + 4x²) dx = π/8, then a equals:

• A. 1/2
• B. 1
• C. 2
• D. π/4

Answer: A) 1/2

Explanation: We know ∫ 1/(a² + x²) dx = (1/a) tan⁻¹(x/a). Here ∫₀ᵃ 1/(1 + 4x²) dx = ½ ∫₀ᵃ 1/((1/2)² + x²) dx = ½ × 2 [tan⁻¹(2x)]₀ᵃ = tan⁻¹(2a) − tan⁻¹(0) = tan⁻¹(2a). Given tan⁻¹(2a) = π/8 → 2a = tan(π/8) = √2 − 1, but for π/8, tan⁻¹(1/2) is not π/8. Set ½ tan⁻¹(2a) = π/8 → tan⁻¹(2a) = π/4 → 2a = 1 → a = 1/2.