1. **Problem:** Evaluate the integral $$\int t e^{-2t} \, dt$$ 2. **Formula and method:** Use integration by parts where $$u = t$$ and $$dv = e^{-2t} dt$$. 3. **Calculate derivatives and integrals:** $$du = dt$$ $$v = \int e^{-2t} dt = \frac{e^{-2t}}{-2} = -\frac{1}{2} e^{-2t}$$ 4. **Apply integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$ 5. **Substitute:** $$\int t e^{-2t} dt = t \left(-\frac{1}{2} e^{-2t}\right) - \int \left(-\frac{1}{2} e^{-2t}\right) dt$$ 6. **Simplify:** $$= -\frac{t}{2} e^{-2t} + \frac{1}{2} \int e^{-2t} dt$$ 7. **Integrate remaining integral:** $$\int e^{-2t} dt = -\frac{1}{2} e^{-2t}$$ 8. **Substitute back:** $$= -\frac{t}{2} e^{-2t} + \frac{1}{2} \left(-\frac{1}{2} e^{-2t}\right) + C = -\frac{t}{2} e^{-2t} - \frac{1}{4} e^{-2t} + C$$ 9. **Final answer:** $$\boxed{-\frac{t}{2} e^{-2t} - \frac{1}{4} e^{-2t} + C}$$