1. **State the problem:** We need to find the integral $$\int \sin(2x) e^{2x} \, dx$$. 2. **Formula and method:** This is an integral of the form $$\int e^{ax} \sin(bx) \, dx$$. The formula for this type of integral is: $$\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C$$ where $a$ and $b$ are constants. 3. **Identify constants:** Here, $a = 2$ and $b = 2$. 4. **Apply the formula:** $$\int \sin(2x) e^{2x} \, dx = \frac{e^{2x}}{2^2 + 2^2} (2 \sin(2x) - 2 \cos(2x)) + C$$ 5. **Simplify denominator:** $$2^2 + 2^2 = 4 + 4 = 8$$ 6. **Final answer:** $$\int \sin(2x) e^{2x} \, dx = \frac{e^{2x}}{8} (2 \sin(2x) - 2 \cos(2x)) + C = \frac{e^{2x}}{4} (\sin(2x) - \cos(2x)) + C$$ This is the integral of the given function.