Class 12 Integrals · Very Long Answer5 Marksdefinite_integration
Class 12 Maths · Integrals · Very Long Answer #177
Evaluate the convergent improper integral \(\int_0^{\pi/4}dx/[\cos^3x\sqrt{2\sin2x}]\).
✅ Step-by-Step Solution
Final Answer: 6/5
- For 0<x≤π/4, √(2sin2x)=2cos x√(tan x), since cosine is positive.
- Thus I=(1/2)∫₀^(π/4)sec⁴x/√(tan x)dx. Set u=tan x, du=sec²x dx; the limits become 0,1.
- Using sec²x=1+u², the transformed integral is (1/2)∫₀¹[u^(−1/2)+u^(3/2)]du. Both powers are integrable at zero.
- Its primitive is √u+u^(5/2)/5. Evaluate the limiting lower endpoint 0 and upper endpoint 1.
- Therefore the integral equals 6/5.
📋 CBSE Stepwise Marking Scheme
- [1 mark] For 0<x≤π/4, √(2sin2x)=2cos x√(tan x), since cosine is positive.
- [1 mark] Thus I=(1/2)∫₀^(π/4)sec⁴x/√(tan x)dx. Set u=tan x, du=sec²x dx; the limits become 0,1.
- [1 mark] Using sec²x=1+u², the transformed integral is (1/2)∫₀¹[u^(−1/2)+u^(3/2)]du. Both powers are integrable at zero.
- [1 mark] Its primitive is √u+u^(5/2)/5. Evaluate the limiting lower endpoint 0 and upper endpoint 1.
- [1 mark] Therefore the integral equals 6/5.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).