Answer
$$ - \frac{1}{3}{\cosh ^{ - 1}}\left( {\frac{{3x}}{5}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{\sqrt {9{x^2} - 25} }}} \cr
& or \cr
& = \int {\frac{{dx}}{{\sqrt {{{\left( {3x} \right)}^2} - {{\left( 5 \right)}^2}} }}} \cr
& {\text{substitute }}u = 3x,{\text{ }}du = 3dx,{\text{ }}a = 5 \cr
& = \int {\frac{{dx}}{{\sqrt {{{\left( {3x} \right)}^2} - {{\left( 5 \right)}^2}} }}} = \int {\frac{{\left( {1/3} \right)du}}{{\sqrt {{u^2} - {a^2}} }}} \cr
& = \frac{1}{3}\int {\frac{{du}}{{\sqrt {{u^2} - {a^2}} }}} \cr
& {\text{find the antiderivarive using the theorem 6}}{\text{.8}}{\text{.6 }}\left( {{\text{see page 480}}} \right) \cr
& = - \frac{1}{3}{\cosh ^{ - 1}}\left( {\frac{u}{a}} \right) + C \cr
& {\text{write in terms of }}x \cr
& = - \frac{1}{3}{\cosh ^{ - 1}}\left( {\frac{{3x}}{5}} \right) + C \cr} $$
