Mathbee CBSE Mathbee · CBSE Class 12 Maths
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q293 Marksintegrals

CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Question 29 (Integrals)

If I₁ = ∫₋π/4^π/4 dx/(1+cos2x) and I₂ = ∫₋1/2^1/2 |x| dx, then show that I₁ − 4I₂ = 0.

✅ Step-by-Step Solution

Final Answer: I₁ − 4I₂ = 0
  1. I₁ = 2∫₀^{π/4} dx/(2cos²x) = ∫₀^{π/4} sec²x dx = [tan x]₀^{π/4} = 1.
  2. I₂ = 2∫₀^{1/2} x dx = 2 [x²/2]₀^{1/2} = (1/2)² = 1/4. I₁ − 4I₂ = 1 − 1 = 0.
  3. I₁ − 4I₂ = 0

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] I₁ = 2∫₀^{π/4} dx/(2cos²x) = ∫₀^{π/4} sec²x dx = [tan x]₀^{π/4} = 1.
  • [2 marks] I₂ = 2∫₀^{1/2} x dx = 2 [x²/2]₀^{1/2} = (1/2)² = 1/4. I₁ − 4I₂ = 1 − 1 = 0.
  • [0.5 marks] I₁ − 4I₂ = 0

Master Chapter 7: Integrals

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
📘 Chapter Notes & All Exercises 🚀 Open in Interactive Workspace →
← Previous Question
Find: ∫ √((x+2)/(x−2)) dx. OR Find: ∫ x²/((x²+9) (x²+16)) dx.
Next Question →
Solve \((y^2-x^2)\,dx=2xy\,dy\). OR Solve…

All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/1/3

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38

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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).