Let $x = a \sec \theta \; \; \; \implies dx = a \; \sec \theta \; \tan \theta \; d \theta$
$ \begin{align*}\int \frac{dx}{\sqrt {x^2 - a^2}} &= \int \frac {a \; \sec \theta \; \tan \theta \; d \theta}{\sqrt{a \sec^2 \theta - a^2}} \\ \int \sec \theta \; d \theta &= log | \sec \theta + \tan \theta \\ & = log \begin{vmatrix} \frac{x}{a} + \sqrt{ \frac{x^2}{a^2} + 1} \end{vmatrix} +C \\ & = log |x + \sqrt {x^2 - a^2} | - log |a| + C \\ & = log |x + \sqrt{x^2 - a^2}| +A\end{align*}$