CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q313 Marksdefinite_integral_symmetry
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Question 31 (Integrals)
Evaluate \(\int_{\pi/12}^{5\pi/12}\frac{dx}{1+\sqrt{\cot x}}\). OR Evaluate \(\int_{-\pi/6}^{\pi/2}(\sin|x|+\cos|x|)\,dx\).
Alternative 1 3 Marks
Evaluate \(\int_{\pi/12}^{5\pi/12}\frac{dx}{1+\sqrt{\cot x}}\).
Answer: π/6.
- The substitution x↦π/2−x swaps the endpoints and replaces cot x with tan x.
- Add the original and reflected integrands. Since √tan x=1/√cot x, their sum is 1.
- Twice the integral is the interval length π/3, so the integral equals π/6.
Alternative 2 3 Marks
Evaluate \(\int_{-\pi/6}^{\pi/2}(\sin|x|+\cos|x|)\,dx\).
Answer: (7−√3)/2.
- Split at zero and reflect the negative interval to [0,π/6].
- On both positive intervals the primitive is −cos x+sin x. The contributions are (3−√3)/2 and 2.
- Their sum is (7−√3)/2.