CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1 · Q325 Markssubstitution
CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/1 · Question 32 (Integrals)
Evaluate \(\int_{\pi/6}^{\pi/3}(\sin x+\cos x)/\sqrt{\sin2x}\,dx\).
✅ Step-by-Step Solution
Final Answer: \(2\sin^{-1}((\sqrt3-1)/2)\)
- Set u=sin x−cos x, so du=(cos x+sin x)dx.
- Since sin2x=1−u², the primitive is sin⁻¹u. The limits become (1−√3)/2 and (√3−1)/2, both in (−1,1).
- Arcsine is odd, so the endpoint difference is 2 sin⁻¹((√3−1)/2).
📋 CBSE Stepwise Marking Scheme
- [1 mark] Set u=sin x−cos x, so du=(cos x+sin x)dx.
- [3 marks] Since sin2x=1−u², the primitive is sin⁻¹u. The limits become (1−√3)/2 and (√3−1)/2, both in (−1,1).
- [1 mark] Arcsine is odd, so the endpoint difference is 2 sin⁻¹((√3−1)/2).
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