Answer
$$\frac{1}{3}\sin \left( {3{e^t} - 2} \right) + C $$
Work Step by Step
$$\eqalign{
& \int {{e^t}\cos \left( {3{e^t} - 2} \right)} dt \cr
& {\text{integrate by the substitution method}} \cr
& {\text{set }}u = 3{e^t} - 2{\text{ then }}\frac{{du}}{{dt}} = 3{e^t},\,\,\,\,dt = \frac{{du}}{{3{e^t}}} \cr
& {\text{write the integrand in terms of }}u \cr
& \int {{e^t}\cos \left( {3{e^t} - 2} \right)} dt = \int {{e^t}\cos u} \left( {\frac{{du}}{{3{e^t}}}} \right) \cr
& {\text{cancel common terms}} \cr
& = \int {\cos u} \left( {\frac{{du}}{3}} \right) \cr
& = \frac{1}{3}\int {\cos u} du \cr
& {\text{integrating}} \cr
& = \frac{1}{3}\sin u + C \cr
& {\text{replace }}3{e^t} - 2{\text{ for }}u \cr
& = \frac{1}{3}\sin \left( {3{e^t} - 2} \right) + C \cr} $$
