Question

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

• A. $2\, log\, \left|\sqrt{x}-1\right|+C$
• B. $2\, log\, \left|\sqrt{x}+1\right|+C$
• 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 $I=\int \frac{d x}{x-\sqrt{x}}=\int \frac{d x}{\sqrt{x}(\sqrt{x}-1)}$
Put $\sqrt{x}-1=t$
$\Rightarrow \frac{1}{2 \sqrt{x}} d x=d t$
$\therefore I=\int \frac{2 d t}{t}=2 \log |t|+C$
$=2 \log |\sqrt{x}-1|+C$