Answer
$${\sin ^{ - 1}}\left( {\frac{{\sinh x}}{3}} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\cosh x}}{{\sqrt {9 - {{\sinh }^2}x} }}} dx \cr
& {\text{Let }}u = \sinh x,{\text{ }}du = \cosh xdx \cr
& \int {\frac{{\cosh x}}{{\sqrt {9 - {{\sinh }^2}x} }}} dx = \int {\frac{{du}}{{\sqrt {9 - {u^2}} }}} \cr
& {\text{Use the formula }}\int {\frac{{du}}{{\sqrt {{a^2} - {u^2}} }} = {{\sin }^{ - 1}}\left( {\frac{u}{a}} \right) + C} \cr
& \int {\frac{{du}}{{\sqrt {9 - {u^2}} }}} = {\sin ^{ - 1}}\left( {\frac{u}{3}} \right) + C \cr
& {\text{Write in terms of }}x \cr
& = {\sin ^{ - 1}}\left( {\frac{{\sinh x}}{3}} \right) + C \cr} $$
