CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q273 Marksdefinite_integral_symmetry
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Question 27 (Integrals)
Evaluate \(\int_0^\pi\frac{\sin^{2026}x}{\sin^{2026}x+\cos^{2026}x}\,dx\).
✅ Step-by-Step Solution
Final Answer: π/2.
- Reflection around π/2 leaves the integrand unchanged since the exponent is even. Thus the integral is twice that on [0,π/2].
- On [0,π/2], reflect x↦π/2−x to interchange sine and cosine. The original and reflected fractions sum to 1.
- The half-interval integral is π/4, so the requested value is π/2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Reflection around π/2 leaves the integrand unchanged since the exponent is even. Thus the integral is twice that on [0,π/2].
- [2 marks] On [0,π/2], reflect x↦π/2−x to interchange sine and cosine. The original and reflected fractions sum to 1.
- [0.5 marks] The half-interval integral is π/4, so the requested value is π/2.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int\frac{dx}{x^{1/2}+x^{1/3}}\). OR Evaluate…
Next Question →Evaluate \(\int\frac{\cos x\,dx}{(2+\sin x)(4+\sin x)}\). OR…
All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/3/1
Related Questions in Integrals
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).