CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q313 Marksdefinite_integral_symmetry
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Question 31 (Integrals)
Evaluate \(\int_{-\pi/2}^{\pi/2}\frac{\cos^2x}{2^x+1}\,dx\).
✅ Step-by-Step Solution
Final Answer: π/4.
- Reflect x to −x, leaving cos²x unchanged.
- The two reciprocal denominators add to 1 because 1/(1+2ˣ)+1/(1+2⁻ˣ)=1. Thus twice the requested integral is the integral of cos²x over the symmetric interval.
- That integral is π/2, so the requested value is π/4.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Reflect x to −x, leaving cos²x unchanged.
- [2 marks] The two reciprocal denominators add to 1 because 1/(1+2ˣ)+1/(1+2⁻ˣ)=1. Thus twice the requested integral is the integral of cos²x over the symmetric interval.
- [0.5 marks] That integral is π/2, so the requested value is π/4.
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