Question
Integrate the function: $\frac{\sqrt{\tan x}}{\sin x \cos x}$
✓
Answer
Let $I=\int \frac{\sqrt{\tan x}}{\sin x \cos x}$
$=\int \frac{\sqrt{\tan x} \cdot \cos x}{\sin x \cos x \cdot \cos x} d x$
$= \int \frac{\sqrt{\tan x}}{\tan x \cos ^{2} x} d x$
$= \int \frac{\sec ^{2} x d x}{\sqrt{\tan x}}$
Let tan x = t $\Rightarrow$ sec2x dx = dt
$\Rightarrow I=\int \frac{d t}{\sqrt{t}}$
$=2 \sqrt{t}+C$
$=2 \sqrt{\tan x}+c$
$=\int \frac{\sqrt{\tan x} \cdot \cos x}{\sin x \cos x \cdot \cos x} d x$
$= \int \frac{\sqrt{\tan x}}{\tan x \cos ^{2} x} d x$
$= \int \frac{\sec ^{2} x d x}{\sqrt{\tan x}}$
Let tan x = t $\Rightarrow$ sec2x dx = dt
$\Rightarrow I=\int \frac{d t}{\sqrt{t}}$
$=2 \sqrt{t}+C$
$=2 \sqrt{\tan x}+c$
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