Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #93
Evaluate \(\int_0^\pi4x\sin x/(1+\cos^2x)dx\).
✅ Step-by-Step Solution
Final Answer: π²
- Let I be the integral. Reflect x↦π−x; sine and cos² are unchanged, giving I=∫₀^π4(π−x)sin x/(1+cos²x)dx.
- Add the equations: 2I=4π∫₀^πsin x/(1+cos²x)dx.
- With u=cos x, the remaining integral is ∫₋₁¹du/(1+u²)=π/2, so I=π².
- Therefore the integral equals π².
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let I be the integral. Reflect x↦π−x; sine and cos² are unchanged, giving I=∫₀^π4(π−x)sin x/(1+cos²x)dx.
- [1 mark] Add the equations: 2I=4π∫₀^πsin x/(1+cos²x)dx.
- [1 mark] With u=cos x, the remaining integral is ∫₋₁¹du/(1+u²)=π/2, so I=π².
- [0.5 marks] Therefore the integral equals π².
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