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Grade 12th passAlgebra

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

John Doe

12 Years agoGrade 12th pass

1 Answer

Jitender Singh

12 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

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