1. **State the problem:** We need to find the integral $$\int e^{2x} \cos(3x) \, dx$$. 2. **Formula and method:** This is an integral of the form $$\int e^{ax} \cos(bx) \, dx$$. The formula for this type of integral is: $$\int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos(bx) + b \sin(bx)) + C$$ where $a$ and $b$ are constants. 3. **Identify constants:** Here, $a = 2$ and $b = 3$. 4. **Apply the formula:** $$\int e^{2x} \cos(3x) \, dx = \frac{e^{2x}}{2^2 + 3^2} (2 \cos(3x) + 3 \sin(3x)) + C$$ 5. **Simplify the denominator:** $$2^2 + 3^2 = 4 + 9 = 13$$ 6. **Final answer:** $$\int e^{2x} \cos(3x) \, dx = \frac{e^{2x}}{13} (2 \cos(3x) + 3 \sin(3x)) + C$$ This formula works because the integral of the product of an exponential and a trigonometric function can be solved using integration by parts twice or by recognizing the pattern and using the formula directly.