Answer
$$\frac{1}{13}e^{2 x}(2 \cos 3 x+3 \sin 3 x)+C$$
Work Step by Step
Given
$$\int e^{2 x} \cos 3 x d x$$
Use the rule
$$\int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}+C$$
Here $a= 2,\ b= 3 $ ,then
\begin{aligned}
\int e^{2x} \cos 3 x d x&=\frac{e^{2 x}(2 \cos 3 x+3 \sin 3 x)}{2^{2}+3^{2}}+C\\
&= \frac{1}{13}e^{2 x}(2 \cos 3 x+3 \sin 3 x)+C\\
\end{aligned}
