Mathematics

If $$\displaystyle I=\int { \left( \sqrt { \tan { x }  } +\sqrt { \cot { x }  }  \right)  } dx=f\left( x \right)+c$$


ANSWER

$$\sqrt { 2 } \sin ^{ -1 }\times \left( \sin { x } -\cos { x } \right) $$


SOLUTION
Given,

$$\displaystyle \int \sqrt{\tan x}+\sqrt{\cot x}dx$$

$$\displaystyle \int \dfrac{\sin x+\cos x}{\sqrt{\sin x\cos x}}dx$$

put $$\sin x-\cos x=u\rightarrow du=\sin x+\cos x dx$$

$$u^2=\sin^2 x+\cos^2 x-2\sin x\cos x\Rightarrow \sin x\cos x=\dfrac {1-u^2}{2}$$

$$\displaystyle I=\int \dfrac{\sqrt{2}du}{\sqrt{1-u^2}}$$

$$=\sqrt{2}\sin ^{-1}u+c$$

$$=\sqrt{2}\sin^{-1}(\sin x-\cos x)+c$$
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Single Correct Medium Published on 17th 09, 2020
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