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

The correct option is B $c-\frac{1}{a}{\sin }^{-1}\frac{a}{|x|}$

$\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}{\sec }^{-1}\frac{|x|}{a}=\frac{1}{a}(\frac{\pi }{2}-{\sin }^{-1}\frac{a}{|x|})$
$=c-\frac{1}{a}{\sin }^{-1}\frac{a}{|x|}$, where $c=\frac{\pi }{2a}$