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