Step 1: Set up the integral for the given expression. We need to integrate $\sin(3x) - e^{3x}$ with respect to $x$. $$ \int (\sin(3x) - e^{3x})\,dx $$ Step 2: Integrate each term separately. The integral of a difference is the difference of the integrals. $$ \int \sin(3x)\,dx - \int e^{3x}\,dx $$ For the first term, $\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax) + C$. Here $a=3$. $$ \int \sin(3x)\,dx = -\frac{1}{3}\cos(3x) $$ For the second term, $\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C$. Here $a=3$. $$ \int e^{3x}\,dx = \frac{1}{3}e^{3x} $$ Step 3: Combine the results and add the constant of integration. $$ -\frac{1}{3}\cos(3x) - \frac{1}{3}e^{3x} + C $$ The final answer is $\boxed{-\frac{1}{3}\cos(3x) - \frac{1}{3}e^{3x} + C}$.