Answer
$$\int e^{2t}\cos3tdt=\frac{e^{2t}}{13}(2\cos3t+3\sin3t)+C$$
Work Step by Step
$$A=\int e^{2t}\cos3tdt$$
Use Formula 115, which states that $$\int e^{ax}\cos bxdx=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C$$
for $a=2$ and $b=3$ here:
$$A=\frac{e^{2t}}{2^2+3^2}(2\cos3t+3\sin3t)+C$$ $$A=\frac{e^{2t}}{13}(2\cos3t+3\sin3t)+C$$