Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #88
Evaluate the convergent improper integral \(\int_0^{\pi/2}[\ln(\sin x)-\ln(2\cos x)]dx\).
✅ Step-by-Step Solution
Final Answer: −(π/2)ln2
- Each endpoint logarithmic singularity is integrable. Write I=J−(π/2)ln2, where J=∫₀^(π/2)ln(tan x)dx.
- Reflect x↦π/2−x. Then J=∫ln(cot x)dx=−J because tan x>0 in the interior.
- Hence J=0 and I=−(π/2)ln2.
- Therefore the integral equals −(π/2)ln2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Each endpoint logarithmic singularity is integrable. Write I=J−(π/2)ln2, where J=∫₀^(π/2)ln(tan x)dx.
- [1 mark] Reflect x↦π/2−x. Then J=∫ln(cot x)dx=−J because tan x>0 in the interior.
- [1 mark] Hence J=0 and I=−(π/2)ln2.
- [0.5 marks] Therefore the integral equals −(π/2)ln2.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int_0^{2\pi}dx/(1+e^{\sin x})\).
Next Question →Evaluate \(\int_0^{\pi/2}e^x\sin x\,dx\).
More Practice Questions in Integrals
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).