Class 12 Integrals · Very Long Answer5 Marksdefinite_integration
Class 12 Maths · Integrals · Very Long Answer #114
Evaluate \(\int_0^{\pi/2}\sin^2x/(\sin x+\cos x)dx\).
✅ Step-by-Step Solution
Final Answer: ln(√2+1)/√2
- Let I be the integral. Reflection x↦π/2−x gives I=∫₀^(π/2)cos²x/(sin x+cos x)dx.
- Adding gives 2I=∫₀^(π/2)dx/(sin x+cos x).
- Use sin x+cos x=√2 cos(x−π/4) and u=x−π/4, with limits −π/4,π/4.
- Thus 2I=(1/√2)[ln|sec u+tan u|]₋π/₄^π/₄=(1/√2)ln[(√2+1)/(√2−1)]=√2ln(√2+1).
- Therefore the integral equals ln(√2+1)/√2.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Let I be the integral. Reflection x↦π/2−x gives I=∫₀^(π/2)cos²x/(sin x+cos x)dx.
- [1 mark] Adding gives 2I=∫₀^(π/2)dx/(sin x+cos x).
- [1 mark] Use sin x+cos x=√2 cos(x−π/4) and u=x−π/4, with limits −π/4,π/4.
- [1 mark] Thus 2I=(1/√2)[ln|sec u+tan u|]₋π/₄^π/₄=(1/√2)ln[(√2+1)/(√2−1)]=√2ln(√2+1).
- [1 mark] Therefore the integral equals ln(√2+1)/√2.
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