Evaluate $\int \frac{1}{\sqrt{9 x^{2} + 1}}\,dx$.

Explanation

1. 01

   Put $t = 3x$, $dt = 3\,dx$: $\frac{1}{3}\int \frac{dt}{\sqrt{t^{2} + 1}}$.

2. 02

   $= \frac{1}{3}\log\left|t + \sqrt{t^{2} + 1}\right| = \frac{1}{3}\log\left|3x + \sqrt{9x^{2} + 1}\right| + C$.