Mathbee CBSE Mathbee · CBSE Class 12 Maths
Class 12 Integrals · Long Answer3 Marksimproper_integral_divergence

Class 12 Maths · Integrals · Long Answer #171

Evaluate as an improper integral, or establish divergence: \(\int_{-\pi/2}^{\pi/2}e^{2x}(1-\sin2x)/(1-\cos2x)dx\).

✅ Step-by-Step Solution

Final Answer: Diverges to +∞.
  1. The denominator vanishes at the interior point x=0. Any improper integral must be split into separate one-sided integrals there.
  2. As x→0, e^(2x)→1, 1−sin2x→1 and 1−cos2x∼2x². Thus the integrand is asymptotic to 1/(2x²)>0.
  3. The integral of 1/x² diverges at 0, so comparison shows both one-sided contributions diverge to +∞.
  4. The stated improper integral therefore diverges to +∞; it has no finite value.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] The denominator vanishes at the interior point x=0. Any improper integral must be split into separate one-sided integrals there.
  • [1 mark] As x→0, e^(2x)→1, 1−sin2x→1 and 1−cos2x∼2x². Thus the integrand is asymptotic to 1/(2x²)>0.
  • [1 mark] The integral of 1/x² diverges at 0, so comparison shows both one-sided contributions diverge to +∞.
  • [0.5 marks] The stated improper integral therefore diverges to +∞; it has no finite value.

Master Chapter 7: Integrals

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
📘 Chapter Notes & All Exercises 🚀 Open Interactive Workspace →
← Previous Question
Evaluate \(\int_{-\pi}^{\pi}2x(1+\sin x)/(1+\cos^2x)dx\).
Next Question →
Evaluate the convergent improper integral \(\int_0^{\pi/2}\ln(\sin…

More Practice Questions in Integrals

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38Q39Q40

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