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Integrate the functions  tan x/ sin x cos x

Q34  Integrate the functions  \frac{\sqrt { \tan x } }{\sin x \cos x }

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Given function  \frac{\sqrt { \tan x } }{\sin x \cos x }

Assume that I = \int \frac{\sqrt { \tan x } }{\sin x \cos x }dx

 Now solving the assumed integral;

Multiplying numerator and denominator by \cos x;

I = \int \frac{\sqrt{\tan x }\times\cos x}{\sin x \cos x\times \cos x}dx

= \int \frac{\sqrt{\tan x }}{\tan x \cos^2 x } dx

= \int \frac{\sec^2 x }{\sqrt{\tan x }}dx

Now, to solve further we will assume \tan x =t

Or, \sec^2{x}dx =dt

\therefore I =\int \frac{dt}{\sqrt t}

=2\sqrt t +C

Now, back substituting the value of t,

= 2\sqrt{\tan x } +C 

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