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Evaluate the following integral ∫(cos2x)^3/2 .cosx dx integral limit is from 0 to pi/4

Evaluate the following integral
 
∫(cos2x)^3/2 .cosx dx
integral limit is from 0 to pi/4

Grade:12th pass

2 Answers

Saurabh Koranglekar
askIITians Faculty 10341 Points
2 years ago
576-225_1.PNG576-775_2.PNG576-2291_3.PNG576-2441_4.PNG
Prakhar Agarwal
26 Points
10 months ago
∫cosx.(cos2x)3/2dx=∫(1-sin2x)3/2cosxdx
Limits --> 0 to π/4
Put u=sinx,
=∫(1-2u2)3/2du
Limits --> 0 to 1/√2
Put u=sint/√2,
=1/√2∫cost(1-sin2t)3/2dt
Limits --> 0 to π/2
=1/√2∫cos4tdt=E(let)
Limits --> same as above
2E=1/√2∫(sin4t+cos4t)dt    (Applying integration property)
E=1/2√2∫(1-2sin2t.cos2t)dt
=1/2√2∫(1-(sin22t)/2)dt
=1/8√2∫(3+cos4t)dt
Limits --> 0 to π/2
After integrating, (now it's simple)
E=3π/16√2       (Ans.)

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