Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #167
Evaluate \(\int_0^\pi x/(1+\sin x)dx\).
✅ Step-by-Step Solution
Final Answer: π
- Let g(x)=1/(1+sin x), symmetric under x↦π−x. Reflection gives I=(π/2)∫₀^πg(x)dx.
- A nonsingular primitive of g on [0,π] is −cos x/(1+sin x), since its derivative is 1/(1+sin x).
- The endpoint difference is 1−(−1)=2, hence I=π.
- Therefore the integral equals π.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let g(x)=1/(1+sin x), symmetric under x↦π−x. Reflection gives I=(π/2)∫₀^πg(x)dx.
- [1 mark] A nonsingular primitive of g on [0,π] is −cos x/(1+sin x), since its derivative is 1/(1+sin x).
- [1 mark] The endpoint difference is 1−(−1)=2, hence I=π.
- [0.5 marks] Therefore the integral equals π.
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