1. **State the problem:** We need to evaluate the integral $$\int e^{-4x} \cos 3x \, dx$$. 2. **Formula and method:** For integrals of the form $$\int e^{ax} \cos(bx) \, dx$$, we use integration by parts or recognize the standard formula: $$\int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos bx + b \sin bx) + C$$ 3. **Identify constants:** Here, $$a = -4$$ and $$b = 3$$. 4. **Apply the formula:** $$\int e^{-4x} \cos 3x \, dx = \frac{e^{-4x}}{(-4)^2 + 3^2} (-4 \cos 3x + 3 \sin 3x) + C$$ 5. **Calculate denominator:** $$(-4)^2 + 3^2 = 16 + 9 = 25$$ 6. **Final answer:** $$\int e^{-4x} \cos 3x \, dx = \frac{e^{-4x}}{25} (-4 \cos 3x + 3 \sin 3x) + C$$