Question

What is $\int \frac{dx}{\sqrt{x^2-a^2}}$ equal to?
where $c$ is the constant of integration.

• A. $\ln \left|\frac{x+\sqrt{x^2+a^2}}{a}\right|+c$
• B. $\ln \left|\frac{x-\sqrt{x^2+a^2}}{a}\right|+c$
• C. $\ln \left|\frac{x^2+\sqrt{x^2+a^2}}{a}\right|+c$
• D. None of the above

Answer 1

The correct option is D None of the above

Let $I=\int \frac{dx}{\sqrt{x^2-a^2}}$

Also, let $x=a\sec \theta ⟹dx=a\sec \theta \tan \theta d\theta$

$\therefore I=\int \frac{a\sec \theta \tan \theta d\theta }{\sqrt{a^2({\sec }^2\theta -1)}}$

$\therefore I=\int \frac{a\sec \theta \tan \theta d\theta }{\sqrt{a^2{\tan }^2\theta }}$

$\therefore I=\int \frac{a\sec \theta \tan \theta d\theta }{a\tan \theta }$

$\therefore I=\int \sec \theta d\theta$

$\therefore I=\ln |\sec x+\tan x|+c$

$\therefore I=\ln \left|\frac{a+\sqrt{x^2-a^2}}{a}\right|+c$