int_0^{pi / 4} frac{d x}{cos ^3(x) cdot sqrt{ | vedclass

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(A) Let $I = \int_0^{\pi / 4} \frac{d x}{\cos ^3 x \sqrt{2 \sin 2 x}}$.Using the identity $\sin 2x = 2 \sin x \cos x$,we get:$I = \int_0^{\pi / 4} \fr

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$\frac{6}{5}$

Vedclass · Answer

(A) Let $I = \int_0^{\pi / 4} \frac{d x}{\cos ^3 x \sqrt{2 \sin 2 x}}$.Using the identity $\sin 2x = 2 \sin x \cos x$,we get:$I = \int_0^{\pi / 4} \frac{d x}{\cos ^3 x \sqrt{4 \sin x \cos x}} = \int_0^{\pi / 4} \frac{d x}{2 \cos ^4 x \sqrt{	an x}}$.$I = \frac{1}{2} \int_0^{\pi / 4} \frac{\sec^2 x (1 + 	an^2 x)}{\sqrt{	an x}} d x$.Let $	an x = t^2$,then $\sec^2 x d x = 2t dt$.When $x = 0, t = 0$ and when $x = \frac{\pi}{4}, t = 1$.Substituting these into the integral:$I = \frac{1}{2} \int_0^1 \frac{(1 + t^4) \cdot 2t dt}{t} = \int_0^1 (1 + t^4) dt$.$I = [t + \frac{t^5}{5}]_0^1 = 1 + \frac{1}{5} = \frac{6}{5}$.

$\int_0^{\pi / 4} \frac{d x}{\cos ^3(x) \cdot \sqrt{2 \sin (2 x)}}=$

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