Question

$\int \frac{dx}{x-\sqrt{x}}$ is equal to :

• A. $2\log |\sqrt{x}-1|+C$
• B. $2\log |\sqrt{x}+1|+C$
• C. $\log |\sqrt{x}-1|+C$
• D. $\frac{1}{2}\log |\sqrt{x}+1|+C$
• E. $\frac{1}{2}\log |\sqrt{x}-1|+C$

Answer 1

The correct option is A $2\log |\sqrt{x}-1|+C$

Let $I=\int \frac{dx}{x-\sqrt{x}}$

$=\int \frac{dx}{\sqrt{x}(\sqrt{x}-1)}$

Put $\sqrt{x}-1=t$
$\Rightarrow \frac{1}{2\sqrt{x}}dx=dt$
Therefore, $I=\int \frac{2dt}{t}=2\log |t|+C$
$=2\log |\sqrt{x}-1|+C$