CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Q293 Marksdefinite_integral_symmetry
CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Question 29 (Integrals)
Evaluate \(\int_0^\pi x\,dx/(1+\sin x)\).
✅ Step-by-Step Solution
Final Answer: π.
- The factor 1/(1+sin x) is invariant under x↦π−x.
- Thus the integral is (π/2) times J=∫₀^π dx/(1+sin x). Using t=tan(x/2), J=∫₀^∞2dt/(t+1)²=2.
- Multiply to obtain π.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The factor 1/(1+sin x) is invariant under x↦π−x.
- [2 marks] Thus the integral is (π/2) times J=∫₀^π dx/(1+sin x). Using t=tan(x/2), J=∫₀^∞2dt/(t+1)²=2.
- [0.5 marks] Multiply to obtain π.
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