Question

Evaluate: $\int \frac{1}{x+\sqrt{x}}dx$

Answer 1

$\int \frac{1}{x+\sqrt{x}}dx$

Put $\sqrt{x}=t\Rightarrow x=t^2\Rightarrow dx=2tdt$

$\therefore \int \frac{dx}{x+\sqrt{x}}=\int \frac{2tdt}{t^2+t}=2\int \frac{1}{t+1}dt$

$=2\log |t+1|+c$

$=2\log (\sqrt{x}+1)+c$.