Answer
$$\ln \left|\frac{x+\sqrt{x^{2}-9}}{3}\right|+C$$
Work Step by Step
Given $$\int \frac{d x}{\sqrt{x^{2}-9}} $$
Let $$x=3\sec \theta ,\ \ \ dx= 3\sec \theta \tan \theta d \theta $$
Then
\begin{aligned}
\int \frac{d x}{\sqrt{x^{2}-9}} &=\int \frac{3 \sec \theta \tan \theta d \theta}{\sqrt{(3 \sec \theta)^{2}-9}} \\
&=\int \frac{3 \sec \theta \tan \theta d \theta}{\sqrt{9 \sec ^{2} \theta-9}} \\
&=\int \frac{3 \sec \theta \tan \theta d \theta}{3 \sqrt{\sec ^{2} \theta-1}} \\
&=\int \frac{\sec \theta \tan \theta d \theta}{\sqrt{\tan ^{2} \theta}} \\
&=\int \sec \theta d \theta \\
&=\ln |\sec \theta+\tan \theta|+C\\
&=\ln \left|\frac{x+\sqrt{x^{2}-9}}{3}\right|+C
\end{aligned}
