Question

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

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

Answer 1

Answer: (A)

Let $I=\int \frac{dx}{x-\sqrt{x}}=\int \frac{dx}{\sqrt{x}(\sqrt{x}-1)}$
Put $\sqrt{x}-1=t$
$\Rightarrow \frac{1}{2\sqrt{x}}dx=dt$
$\therefore I=\int \frac{2dt}{t}=2\log |t|+C$
$=2\log |\sqrt{x}-1|+C$