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Find the value of the Integralupper limit is p/2lower limit is -p/2f(x)=root of( cos x - cos^3 x)dx.

Find the value of the Integralupper limit is p/2lower limit is -p/2f(x)=root of( cos x - cos^3 x)dx.

Grade:12th pass

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 years ago
Ans:
I = \int_{-\pi /2}^{\pi /2}\sqrt{cosx-cos^{3}x} dx
cos x is an even function, we have
I = 2\int_{0}^{\pi /2}\sqrt{cosx-cos^{3}x} dx
I = 2\int_{0}^{\pi /2}\sqrt{cosx(1-cos^{2}x)} dx
I = 2\int_{0}^{\pi /2}\sqrt{cosx.sin^{2}x} dx
I = 2\int_{0}^{\pi /2}\sqrt{cosx}sinx dx
cosx = t
-sinx dx = dt
x = 0\rightarrow t = 1
x = \frac{\pi }{2}\rightarrow t = 0
I = 2\int_{0}^{1}\sqrt{t}dt
I = (\frac{4t^{3/2}}{3})_{0}^{1} = \frac{4}{3}
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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