1. **State the problem:** We need to find the integral $$\int xe^{2x} \, dx$$. 2. **Formula and method:** We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = x \implies du = dx$$ $$dv = e^{2x} dx \implies v = \frac{e^{2x}}{2}$$ 3. **Apply integration by parts:** $$\int xe^{2x} dx = x \cdot \frac{e^{2x}}{2} - \int \frac{e^{2x}}{2} dx$$ 4. **Simplify the remaining integral:** $$\int \frac{e^{2x}}{2} dx = \frac{1}{2} \int e^{2x} dx = \frac{1}{2} \cdot \frac{e^{2x}}{2} = \frac{e^{2x}}{4}$$ 5. **Write the final answer:** $$\int xe^{2x} dx = \frac{x e^{2x}}{2} - \frac{e^{2x}}{4} + C$$ where $C$ is the constant of integration.