Mathematics

$$\displaystyle \int \frac{1}{\sqrt{\left ( x \right )}\left [ \sqrt{\left ( x \right )}+1 \right ]}dx.$$


ANSWER

$$\displaystyle 2\log \left ( 1+\sqrt{x} \right ).$$


SOLUTION
Let $$\displaystyle I=\int \frac{1}{\sqrt{\left ( x \right )}\left [ \sqrt{\left ( x \right )}+1 \right ]}dx.$$
Substitute $$ \displaystyle \sqrt { x } +1=t\Rightarrow \frac { 1 }{ 2\sqrt { x }  } dx=dt$$
Therefore 
$$ \displaystyle I=\int  \frac { 2dt }{ t } =2\log  t=2\log  \left( 1+\sqrt { x }  \right) $$
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Single Correct Medium Published on 17th 09, 2020
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