Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #170
Evaluate \(\int_{-\pi}^{\pi}2x(1+\sin x)/(1+\cos^2x)dx\).
✅ Step-by-Step Solution
Final Answer: π²
- Expand into 2x/(1+cos²x)+2x sin x/(1+cos²x). The first term is odd and integrates to zero.
- The second is even, so I=4K with K=∫₀^πx sin x/(1+cos²x)dx. Reflection gives K=(π/2)∫₀^πsin x/(1+cos²x)dx.
- The last integral is ∫₋₁¹du/(1+u²)=π/2. Hence K=π²/4 and I=π².
- Therefore the integral equals π².
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Expand into 2x/(1+cos²x)+2x sin x/(1+cos²x). The first term is odd and integrates to zero.
- [1 mark] The second is even, so I=4K with K=∫₀^πx sin x/(1+cos²x)dx. Reflection gives K=(π/2)∫₀^πsin x/(1+cos²x)dx.
- [1 mark] The last integral is ∫₋₁¹du/(1+u²)=π/2. Hence K=π²/4 and I=π².
- [0.5 marks] Therefore the integral equals π².
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