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Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration2 Marks
Question
Evaluate: $\int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x} d x$
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Answer

$I=\int \frac{\sqrt{\tan x}}{\sin x \cdot \cos x} dx$
Dividing numerator and denominator by cosx.
$ =\int \frac{\frac{\sqrt{\tan x}}{\cos x}}{\frac{\sin x \cos x}{\cos x}} dx$
$ =\int \frac{\sqrt{\tan x}\left(\frac{1}{\cos x}\right)}{\left(\frac{\sin x}{\cos x}\right) \cdot \cos x} dx$
$ =\int \frac{\sqrt{\tan x}}{\frac{\sin x}{\cos x}}\left(\frac{1}{\cos ^2 x}\right) dx$
$ =\int \frac{\sqrt{\tan x}}{\tan x}\left(\frac{1}{\cos ^2 x}\right) dx$
$ =\int \frac{\sqrt{\tan x}}{\tan x}\left(\sec ^2 x\right) dx $
Put, $\tan x = t$
$Sec^2x dx = dt$
$ =\int \frac{1}{\sqrt{t}} d t$
$ =2 \tan ^{\frac{1}{2}}+c$
$ =2 \sqrt{\tan x}+c$

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