Question

The value of the integral $\int \frac{dx}{x\sqrt{x^2-a^2}}$ is equal to:

• A. $c+\frac{1}{a}{\sin }^{-1}\frac{a}{|x|}$
• B. $c-\frac{1}{a}{\sin }^{-1}\frac{a}{|x|}$
• C. $c-\frac{1}{a}{\cos }^{-1}\frac{a}{|x|}$
• D. ${\sin }^{-1}\frac{a}{|x|}+c$

Answer 1

Answer: (B)

Let $I=\int \frac{dx}{x\sqrt{x^2-a^2}}$
Let, $x=\frac{1}{t}$
$\therefore dx=-\frac{1}{t^2}dt$
$\therefore I=\int \frac{-dt}{t^2\cdot \frac{1}{t}\sqrt{{\left(\frac{1}{t^2}\right)}^2-a^2}}=-\frac{1}{a}\int \frac{dt}{\sqrt{{\left(\frac{1}{a}\right)}^2-t^2}}$
$=-\frac{1}{a}{\sin }^{-1}t+C=-\frac{1}{a}{\sin }^{-1}\frac{a}{|x|}+C$
$=C-\frac{1}{a}{\sin }^{-1}\frac{a}{|x|}$