Answer
$${\text{FALSE}}$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{3x\sqrt {9{x^2} - 16} }}} \cr
& {\text{Rewrite the integrand}} \cr
& = \frac{1}{3}\int {\frac{{3dx}}{{\left( {3x} \right)\sqrt {{{\left( {3x} \right)}^2} - {{\left( 4 \right)}^2}} }}} \cr
& {\text{Integate by using the formula }}\int {\frac{{du}}{{u\sqrt {{u^2} - {a^2}} }}} = \frac{1}{a}\operatorname{arcsec} \frac{{\left| u \right|}}{a} + C \cr
& = \frac{1}{3}\left( {\frac{1}{4}\operatorname{arcsec} \frac{{\left| {3x} \right|}}{4}} \right) + C \cr
& = \frac{1}{{12}}\operatorname{arcsec} \frac{{\left| {3x} \right|}}{4} \ne \frac{1}{4}\operatorname{arcsec} \frac{{\left| {3x} \right|}}{4} \cr
& {\text{Therefore, the statement is FALSE}} \cr} $$
