CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q283 Marksdefinite_integral_symmetry
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Question 28 (Integrals)
Evaluate \(\int_0^{\pi/2}\frac{\sin^2x}{1+\sin 2x}\,dx\).
✅ Step-by-Step Solution
Final Answer: 1/2.
- Reflect x to π/2−x, swapping sin²x with cos²x while preserving the denominator. Add to get twice the integral as ∫dx/(1+sin 2x).
- For 0≤x<π/2 put u=tan x. Then dx/(1+sin 2x)=du/(1+u)².
- Its integral from 0 to ∞ is 1, so the original integral equals 1/2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Reflect x to π/2−x, swapping sin²x with cos²x while preserving the denominator. Add to get twice the integral as ∫dx/(1+sin 2x).
- [2 marks] For 0≤x<π/2 put u=tan x. Then dx/(1+sin 2x)=du/(1+u)².
- [0.5 marks] Its integral from 0 to ∞ is 1, so the original integral equals 1/2.
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