Question

Evaluate $\int \frac{dx}{\sqrt{x}(1+\sqrt{x})}$.

• A. $log(1+x)+c$
  
• B. $2log\left(x\right)+c$
• C. $6log(1+\sqrt{x^2})+c$
• D. $2log(1+\sqrt{x})+c$

Answer 1

The correct option is D $2log(1+\sqrt{x})+c$

Consider, $I=\int \frac{dx}{\sqrt{x}(1+\sqrt{x})}$

$1+\sqrt{x}=t$ $\Rightarrow$ $\frac{dx}{2\sqrt{x}}=dt$

$\Rightarrow$ $I=2\int \frac{dt}{t}$

$\Rightarrow$ $I=2\ln \left(t\right)+c$

$\Rightarrow$ $I=2\ln (1+\sqrt{x})+c$.