First slide

Methods of integration

Question

$\int \frac{\sqrt{\tan x}}{\sin x \cos x} d x is equal to$

Moderate

A

$- 2 \sqrt{\tan x} + C$
B

$2 \sqrt{\tan x} + C$
C

$2 \sqrt{\cot x} + C$
D

$- 2 \sqrt{\cot x} + C$

Solution

$Let I = \int \frac{\sqrt{\tan x}}{\sin x \cos x} d x = \int \frac{\sqrt{\tan x}}{\tan x \cos^{2} x} d x$
$I = \int \frac{\sec^{2} x d x}{\sqrt{\tan x}} d x$
$l e t \tan x = t$
$\begin{array}{l} \sec^{2} x d x = d t \\ I = \int \frac{1}{\sqrt{t}} d t = 2 \sqrt{t} + C \\ = 2 \sqrt{\tan x} + C \end{array}$

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