CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q303 Marksintegration_by_parts
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Question 30 (Integrals)
Evaluate \(\int_0^1 x\sin^{-1}x\,dx\).
✅ Step-by-Step Solution
Final Answer: π/8
- Integrate by parts with u=sin⁻¹x and dv=x dx to obtain π/4−(1/2)∫₀¹ x²/√(1−x²) dx.
- For the remaining integral substitute x=sin t, yielding ∫₀^(π/2) sin²t dt=π/4. The endpoint integral converges.
- Therefore the value is π/4−π/8=π/8.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Integrate by parts with u=sin⁻¹x and dv=x dx to obtain π/4−(1/2)∫₀¹ x²/√(1−x²) dx.
- [2 marks] For the remaining integral substitute x=sin t, yielding ∫₀^(π/2) sin²t dt=π/4. The endpoint integral converges.
- [0.5 marks] Therefore the value is π/4−π/8=π/8.
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