Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #91
Evaluate \(\int_{-\pi/2}^{\pi/2}\frac{\sin^{100}x}{\sin^{100}x+\cos^{100}x}dx\).
✅ Step-by-Step Solution
Final Answer: π/2
- The denominator is positive everywhere, and the integrand is even. Thus I=2J where J integrates the same expression from 0 to π/2.
- Reflection x↦π/2−x gives J=∫₀^(π/2)cos¹⁰⁰x/(sin¹⁰⁰x+cos¹⁰⁰x)dx.
- Adding the two forms gives 2J=π/2, hence J=π/4 and I=π/2.
- Therefore the integral equals π/2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] The denominator is positive everywhere, and the integrand is even. Thus I=2J where J integrates the same expression from 0 to π/2.
- [1 mark] Reflection x↦π/2−x gives J=∫₀^(π/2)cos¹⁰⁰x/(sin¹⁰⁰x+cos¹⁰⁰x)dx.
- [1 mark] Adding the two forms gives 2J=π/2, hence J=π/4 and I=π/2.
- [0.5 marks] Therefore the integral equals π/2.
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