Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #95
Evaluate \(\int_0^\pi x\sin x/(1+\cos^2x)dx\).
✅ Step-by-Step Solution
Final Answer: π²/4
- Reflect x↦π−x. The integral also equals ∫₀^π(π−x)sin x/(1+cos²x)dx.
- Adding gives 2I=π∫₀^πsin x/(1+cos²x)dx.
- Let u=cos x. The remaining integral is ∫₋₁¹du/(1+u²)=π/2, so I=π²/4.
- Therefore the integral equals π²/4.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Reflect x↦π−x. The integral also equals ∫₀^π(π−x)sin x/(1+cos²x)dx.
- [1 mark] Adding gives 2I=π∫₀^πsin x/(1+cos²x)dx.
- [1 mark] Let u=cos x. The remaining integral is ∫₋₁¹du/(1+u²)=π/2, so I=π²/4.
- [0.5 marks] Therefore the integral equals π²/4.
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