Evaluate ∫₀¹ x tan⁻¹ x dx.

Evaluate ∫₀¹ x tan⁻¹ x dx.

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

Answer: D) (π − 2)/4

Explanation: Integrate by parts: u = tan⁻¹ x, dv = x dx. du = dx/(1+x²), v = x²/2. ∫ x tan⁻¹ x dx = (x²/2) tan⁻¹ x − (1/2) ∫ x²/(1+x²) dx. x²/(1+x²) = 1 − 1/(1+x²). Integral = x − tan⁻¹ x. So definite: [(x²/2) tan⁻¹ x − (1/2)(x − tan⁻¹ x)]₀¹ = (1/2)(π/4) − (1/2)(1 − π/4) = π/8 − 1/2 + π/8 = π/4 − 1/2 = (π−2)/4.