CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Q325 Marksdefinite_integral_symmetry
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Question 32 (Integrals)
Evaluate \(\int_{-\pi}^{\pi}2x(1+\sin x)/(1+\cos^2x)\,dx\).
✅ Step-by-Step Solution
Final Answer: π².
- The term 2x/(1+cos²x) is odd. The remainder is even, giving 4∫₀^π x sin x/(1+cos²x) dx.
- The factor sin x/(1+cos²x) is invariant under x↦π−x, so replace x by its paired average π/2.
- The result is 2π∫₋₁¹ dt/(1+t²) after t=cos x, giving 2π·π/2=π².
📋 CBSE Stepwise Marking Scheme
- [1 mark] The term 2x/(1+cos²x) is odd. The remainder is even, giving 4∫₀^π x sin x/(1+cos²x) dx.
- [3 marks] The factor sin x/(1+cos²x) is invariant under x↦π−x, so replace x by its paired average π/2.
- [1 mark] The result is 2π∫₋₁¹ dt/(1+t²) after t=cos x, giving 2π·π/2=π².
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