Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #96
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, cos x>0 and √(2sin2x)=2cos x√(tan x). Thus the integrand is sec⁴x/[2√(tan x)].
- Set u=tan x, du=sec²x dx, changing the limits to 0,1. The integral is (1/2)∫₀¹(u^(−1/2)+u^(3/2))du.
- Both powers are integrable at 0. Evaluate [√u+u^(5/2)/5]₀¹=1+1/5.
- Therefore the integral equals 6/5.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] For 0<x≤π/4, cos x>0 and √(2sin2x)=2cos x√(tan x). Thus the integrand is sec⁴x/[2√(tan x)].
- [1 mark] Set u=tan x, du=sec²x dx, changing the limits to 0,1. The integral is (1/2)∫₀¹(u^(−1/2)+u^(3/2))du.
- [1 mark] Both powers are integrable at 0. Evaluate [√u+u^(5/2)/5]₀¹=1+1/5.
- [0.5 marks] Therefore the integral equals 6/5.
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