CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q263 Marksintegrals
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 26 (Integrals)
Evaluate: ∫₀¹ x tan⁻¹ x dx.
✅ Step-by-Step Solution
Final Answer: π/4 − 1/2
- Use integration by parts: u = tan⁻¹x, dv = x dx.
- ∫ = [ (x²/2) tan⁻¹x ]₀¹ − ½∫₀¹ x²/(1+x²) dx. = (π/8) − ½∫₀¹ (1 − 1/(1+x²)) dx. = π/8 − ½ [ x − tan⁻¹x ]₀¹ = π/8 − ½(1 − π/4) = π/8 − 1/2 + π/8 = π/4 − 1/2.
- π/4 − 1/2
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Use integration by parts: u = tan⁻¹x, dv = x dx.
- [2 marks] ∫ = [ (x²/2) tan⁻¹x ]₀¹ − ½∫₀¹ x²/(1+x²) dx. = (π/8) − ½∫₀¹ (1 − 1/(1+x²)) dx. = π/8 − ½ [ x − tan⁻¹x ]₀¹ = π/8 − ½(1 − π/4) = π/8 − 1/2 + π/8 = π/4 − 1/2.
- [0.5 marks] π/4 − 1/2
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