Answer
$$\frac{1}{3}{\sinh ^{ - 1}}\left( {3x} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{\sqrt {1 + 9{x^2}} }}} \cr
& = \int {\frac{{dx}}{{\sqrt {1 + {{\left( {3x} \right)}^2}} }}} \cr
& {\text{substitute }}u = 3x,{\text{ }}du = 3dx \cr
& = \int {\frac{{dx}}{{\sqrt {1 + {{\left( {3x} \right)}^2}} }}} = \frac{1}{3}\int {\frac{{du}}{{\sqrt {1 + {u^2}} }}} \cr
& {\text{find the antiderivarive using the theorem 6}}{\text{.8}}{\text{.6 }}\left( {{\text{see page 480}}} \right) \cr
& = \frac{1}{3}{\sinh ^{ - 1}}u + C \cr
& {\text{write in terms of }}x \cr
& = \frac{1}{3}{\sinh ^{ - 1}}\left( {3x} \right) + C \cr} $$
