Question

Evaluate : $\int \frac{1}{\left(\sqrt{x}+x\right)}dx$

Answer 1

$I=\int \frac{1}{\left(\sqrt{x}+x\right)}dx=\int \frac{1}{\sqrt{x}\left(1+\sqrt{x}\right)}dx$

$Put\left(1+\sqrt{x}\right)=t\frac{1}{\sqrt{x}dx}=2dt$

$\therefore \int \frac{1}{\left(\sqrt{x}+x\right)}dx=\int \frac{1}{\sqrt{x}\left(1+\sqrt{x}\right)}dx=2\int \frac{1}{t}dt=2\log \left|t\right|+C=2\log \left|\left(1+\sqrt{x}\right)\right|+C.$

Hence, this is the answer.