Topic: Engineering Mathematics
$\int\frac{dx}{x\sqrt{x^{2}-a^{2}}}$ is
- A. $\frac{1}{a}cosec^{-1}(\frac{x}{a})$
- B. $\frac{1}{a}sin^{-1}(\frac{x}{a})$
- C. $\frac{1}{a}cos^{-1}(\frac{x}{a})$
- D. $\frac{1}{a}sec^{-1}(\frac{x}{a})$
Answer: D. $\frac{1}{a}sec^{-1}(\frac{x}{a})$
Explanation: Let $x = a \sec \theta$. Then $dx = a \sec \theta \tan \theta d\theta$. Substituting these into the integral gives $\int\frac{a \sec \theta \tan \theta d\theta}{a \sec \theta \sqrt{a^{2} \sec^{2} \theta - a^{2}}} = \int\frac{\tan \theta d\theta}{a \tan \theta} = \int\frac{1}{a}d\theta = \frac{\theta}{a}$. Since $x = a \sec \theta$, $\theta = \sec^{-1}(\frac{x}{a})$. The integral evaluates to $\frac{1}{a}\sec^{-1}(\frac{x}{a})$.