Question

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

• A. $2\log (1-\sqrt{x}).$
• B. $\log (1+\sqrt{x}).$
• C. $\sqrt{2}\log (1+\sqrt{x}).$
• D. $2\log (1+\sqrt{x}).$

Answer 1

Answer: (D)

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

Let $I=\int \frac{1}{\sqrt{\left(x\right)}[\sqrt{\left(x\right)}+1]}dx.$
Substitute $\sqrt{x}+1=t\Rightarrow \frac{1}{2\sqrt{x}}dx=dt$
Therefore
$I=\int \frac{2dt}{t}=2\log t=2\log (1+\sqrt{x})$