If d/dx [F(x)] = 1/(1 + x²), then ∫₀¹ F(x) dx equals:

If d/dx [F(x)] = 1/(1 + x²), then ∫₀¹ F(x) dx equals:

• A. π/4 − 1/2 log 2
• B. π/4
• C. 1/2 log 2
• D. π/2 − log 2

Answer: A) π/4 − 1/2 log 2

Explanation: F(x) = tan⁻¹ x + C. Choose C = 0. Then ∫₀¹ tan⁻¹ x dx. Integrate by parts: u = tan⁻¹ x, dv = dx. ∫ tan⁻¹ x dx = x tan⁻¹ x − ∫ x/(1+x²) dx = x tan⁻¹ x − (1/2) log(1+x²). From 0 to 1: (π/4 − 1/2 log 2) − 0 = π/4 − 1/2 log 2.