tan x sin x cos x ? d x equals:

Question

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

Smart Achievers · Accepted Answer

(1)

Ans. $2 \sqrt{\tan x} + c$

Explanation :

Let $I = \int \frac{\sqrt{\tan x}}{\sin x . \cos x} d x$

$I = \int \frac{\sqrt{\tan x}}{\sin x . \cos x \cdot \frac{\cos x}{\cos x}} d x$

$\Rightarrow$ $I = \int \frac{\sqrt{\tan x}}{\cos^{2} x . (\frac{\sin x}{\cos x})} d x$

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

$I = \int {(\tan x)}^{- 1 / 2} \cdot \frac{1}{\cos^{2} x} d x$

$I = \int {(\tan x)}^{- 1 / 2} \sec^{2} x d x$

Let $\tan x = t$

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

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

$I = \int t^{- 1 / 2} d t$

$I = \frac{t^{1 / 2}}{\frac{1}{2}} + C$

$\Rightarrow$ $I = 2 t^{1 / 2} + C$

$\Rightarrow$ $I = 2 {(\tan x)}^{1 / 2} + C$

$\Rightarrow$ $I = 2 \sqrt{\tan x} + C$

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