1. **State the problem:** Calculate the definite integral $$\int_0^3 x e^{-x} \, dx$$. 2. **Formula and method:** Use integration by parts, where $$\int u \, dv = uv - \int v \, du$$. 3. **Choose parts:** Let $$u = x$$ and $$dv = e^{-x} dx$$. 4. **Compute derivatives and integrals:** Then $$du = dx$$ and $$v = \int e^{-x} dx = -e^{-x}$$. 5. **Apply integration by parts:** $$\int_0^3 x e^{-x} dx = \left. -x e^{-x} \right|_0^3 + \int_0^3 e^{-x} dx$$ 6. **Evaluate the remaining integral:** $$\int_0^3 e^{-x} dx = \left. -e^{-x} \right|_0^3 = -(e^{-3} - 1) = 1 - e^{-3}$$ 7. **Substitute back:** $$-x e^{-x} \bigg|_0^3 = -3 e^{-3} - 0 = -3 e^{-3}$$ 8. **Combine results:** $$\int_0^3 x e^{-x} dx = -3 e^{-3} + 1 - e^{-3} = 1 - 4 e^{-3}$$ 9. **Final answer:** $$\boxed{1 - 4 e^{-3}}$$