First slide

Methods of integration

Question

$\int \frac{dx}{\cos^{3} x \sqrt{2 \sin 2 x}}$ is equal to

Moderate

A

$\sqrt{\tan x} + \frac{\tan^{5 / 2} x}{5} + C$
B

$\sqrt{\tan x} + \frac{2}{5} \tan^{5 / 2} x + C$
C

$2 \sqrt{\tan x} + \frac{2}{5} \tan^{5 / 2} x + C$
D

None of the above

Solution

$Let \begin{array}{l} I = \int \frac{dx}{\cos^{3} x \sqrt{2 \sin 2 x}} \\ = \int \frac{dx}{\cos^{3} x \sqrt{4 \sin xcos x}} = \frac{1}{2} \int \frac{dx}{\cos^{7 / 2} {xsin}^{1 / 2} x} \\ = \frac{1}{2} \int \frac{\sec^{4} x}{\sqrt{\tan x}} dx = \frac{1}{2} \int \frac{(1 + \tan^{2} x) \sec^{2} x}{\sqrt{\tan x}} dx \end{array}$
$\begin{array}{l} Put \tan x = t \Rightarrow \sec^{2} xdx = dt \\ ∴ I = \frac{1}{2} \int \frac{1 + t^{2}}{\sqrt{t}} dt \\ = \frac{1}{2} \int t^{- 1 / 2} dt + \frac{1}{2} \int t^{3 / 2} dt \\ = t^{1 / 2} + \frac{t^{5 / 2}}{5} + C \\ = \sqrt{\tan x} + \frac{1}{5} \tan^{5 / 2} x + C \end{array}$

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