Answer
$$\frac{{{e^{2t}}}}{{13}}\left( {2\cos 3t + 3\sin 3t} \right) + C $$
Work Step by Step
$$\eqalign{
& \int {{e^{2t}}\cos 3t} dt \cr
& {\text{integrate with the table of integrals in the book}} \cr
& {\text{we use formula 115}}:\,\,\,\int {{e^{ax}}\cos bxdx = \frac{{{e^{ax}}}}{{{a^2} + {b^2}}}\left( {a\cos bx + b\sin bx} \right)} + C \cr
& {\text{setting }}a = 2{\text{ and }}b = 3{\text{, then}} \cr
& \int {{e^{2t}}\cos 3t} dt = \frac{{{e^{2t}}}}{{{{\left( 2 \right)}^2} + {{\left( 3 \right)}^2}}}\left( {2\cos 3t + 3\sin 3t} \right) + C \cr
& {\text{simplifying, we get:}} \cr
& = \frac{{{e^{2t}}}}{{13}}\left( {2\cos 3t + 3\sin 3t} \right) + C \cr} $$
