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Evaluate the following integrals:

Question:

Evaluate the following integrals:

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

Solution:

Multiply and divide by $\cos x$

$\Rightarrow \int \frac{\sqrt{\tan x} \cdot \cos x}{\sin x \cdot \cos x \cdot \cos x} d x$

$\Rightarrow \int \frac{\sqrt{\tan x}}{\tan x \cdot \cos ^{2} x} d x$

$\Rightarrow \int \frac{\sec ^{2} x}{\sqrt{\tan x}} d x$

Assume $\tan x=t$

$d(\tan x)=d t$

$\sec ^{2} x d x=d t$

Substituting $t$ and $d t$ in above equation we get

$\Rightarrow \int \frac{1}{\sqrt{t}} \mathrm{dt}$

$\Rightarrow \int \mathrm{t}^{-1 \backslash 2} \cdot \mathrm{dt}$

$\Rightarrow 2 t^{1 / 2}+c$

But $t=\tan x$

$\Rightarrow 2(\tan x)^{1 / 2}+c$

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