1. Stating the problem: Evaluate the integral $$\int x e^{-x} \, dx$$. 2. Formula and method: We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ 3. Choose parts: Let $$u = x$$ so that $$du = dx$$. Let $$dv = e^{-x} dx$$ so that $$v = -e^{-x}$$ (since the integral of $$e^{-x}$$ is $$-e^{-x}$$). 4. Apply integration by parts: $$\int x e^{-x} dx = uv - \int v du = -x e^{-x} - \int (-e^{-x}) dx = -x e^{-x} + \int e^{-x} dx$$ 5. Evaluate the remaining integral: $$\int e^{-x} dx = -e^{-x} + C$$ 6. Combine all parts: $$\int x e^{-x} dx = -x e^{-x} - e^{-x} + C = -(x+1)e^{-x} + C$$ Final answer: $$\boxed{-(x+1)e^{-x} + C}$$