Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #169
Evaluate \(\int_0^\pi x\sin x/(1+3\cos^2x)dx\).
✅ Step-by-Step Solution
Final Answer: π²/(3√3)
- Reflection gives I=(π/2)J, where J=∫₀^πsin x/(1+3cos²x)dx.
- Set u=cos x, obtaining J=∫₋₁¹du/(1+3u²)=[arctan(√3u)/√3]₋₁¹.
- Thus J=2π/(3√3), so I=π²/(3√3)=√3π²/9.
- Therefore the integral equals π²/(3√3).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Reflection gives I=(π/2)J, where J=∫₀^πsin x/(1+3cos²x)dx.
- [1 mark] Set u=cos x, obtaining J=∫₋₁¹du/(1+3u²)=[arctan(√3u)/√3]₋₁¹.
- [1 mark] Thus J=2π/(3√3), so I=π²/(3√3)=√3π²/9.
- [0.5 marks] Therefore the integral equals π²/(3√3).
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