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+1 vote
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in Definite Integrals by (28.8k points)
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Evaluate the following Integral:

\(\int\limits_{0}^{\pi/2}\sqrt{cos\,\text x-cos^3\text x}(sec^2\text x-1) \)cos2x dx

1 Answer

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Best answer

Let I  =  \(\int\limits_{0}^{\pi/2}\sqrt{cos\,\text x-cos^3\text x}(sec^2\text x-1) \)cos2x dx

We have sin2x + cos2x = 1 and sec2x – tan2x = 1

We can write sin3x = sin2x \(\times\) sin x = (1 – cos2x) sin x

Put cos x = t

⇒ –sin(x)dx = dt (Differentiating both sides) 

⇒ sin(x)dx = –dt

When x = 0, t = cos 0 = 1

When x = \(\cfrac{\pi}2\), t = cos \(\cfrac{\pi}2\) = 0

So, the new limits are 1 and 0.

Substituting this in the original integral,

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