Question

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

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

Answer 1

Answer: (A)

$c-\frac{1}{a}{\sin }^{-1}\frac{a}{\left|x\right|}$

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