1. **Stating the problem:** Evaluate the integral $$\int (2x + 1)^3 e^{-x/2} \, dx$$. 2. **Formula and approach:** This integral involves a product of a polynomial and an exponential function. Integration by parts is a suitable method here. Recall the integration by parts formula: $$\int u \, dv = uv - \int v \, du$$ 3. **Choosing parts:** Let $$u = (2x + 1)^3$$ $$dv = e^{-x/2} dx$$ 4. **Compute derivatives and integrals:** $$du = 3(2x + 1)^2 \cdot 2 \, dx = 6(2x + 1)^2 \, dx$$ $$v = \int e^{-x/2} dx = -2 e^{-x/2}$$ 5. **Apply integration by parts:** $$\int (2x + 1)^3 e^{-x/2} dx = u v - \int v \, du = -2 (2x + 1)^3 e^{-x/2} - \int -2 e^{-x/2} \cdot 6 (2x + 1)^2 dx$$ 6. **Simplify the integral:** $$= -2 (2x + 1)^3 e^{-x/2} + 12 \int (2x + 1)^2 e^{-x/2} dx$$ 7. **Repeat integration by parts on $$\int (2x + 1)^2 e^{-x/2} dx$$:** Let $$u = (2x + 1)^2$$ $$dv = e^{-x/2} dx$$ Then $$du = 4(2x + 1) dx$$ $$v = -2 e^{-x/2}$$ 8. **Apply integration by parts again:** $$\int (2x + 1)^2 e^{-x/2} dx = -2 (2x + 1)^2 e^{-x/2} - \int -2 e^{-x/2} \cdot 4 (2x + 1) dx$$ $$= -2 (2x + 1)^2 e^{-x/2} + 8 \int (2x + 1) e^{-x/2} dx$$ 9. **Repeat integration by parts on $$\int (2x + 1) e^{-x/2} dx$$:** Let $$u = (2x + 1)$$ $$dv = e^{-x/2} dx$$ Then $$du = 2 dx$$ $$v = -2 e^{-x/2}$$ 10. **Apply integration by parts:** $$\int (2x + 1) e^{-x/2} dx = -2 (2x + 1) e^{-x/2} - \int -2 e^{-x/2} \cdot 2 dx$$ $$= -2 (2x + 1) e^{-x/2} + 4 \int e^{-x/2} dx$$ 11. **Integrate $$\int e^{-x/2} dx$$:** $$\int e^{-x/2} dx = -2 e^{-x/2} + C$$ 12. **Substitute back:** $$\int (2x + 1) e^{-x/2} dx = -2 (2x + 1) e^{-x/2} + 4 (-2 e^{-x/2}) + C = -2 (2x + 1) e^{-x/2} - 8 e^{-x/2} + C$$ 13. **Substitute into step 8:** $$\int (2x + 1)^2 e^{-x/2} dx = -2 (2x + 1)^2 e^{-x/2} + 8 \left[-2 (2x + 1) e^{-x/2} - 8 e^{-x/2}\right] + C$$ $$= -2 (2x + 1)^2 e^{-x/2} - 16 (2x + 1) e^{-x/2} - 64 e^{-x/2} + C$$ 14. **Substitute into step 6:** $$\int (2x + 1)^3 e^{-x/2} dx = -2 (2x + 1)^3 e^{-x/2} + 12 \left[-2 (2x + 1)^2 e^{-x/2} - 16 (2x + 1) e^{-x/2} - 64 e^{-x/2}\right] + C$$ $$= -2 (2x + 1)^3 e^{-x/2} - 24 (2x + 1)^2 e^{-x/2} - 192 (2x + 1) e^{-x/2} - 768 e^{-x/2} + C$$ 15. **Factor out $$e^{-x/2}$$:** $$= e^{-x/2} \left[-2 (2x + 1)^3 - 24 (2x + 1)^2 - 192 (2x + 1) - 768\right] + C$$ **Final answer:** $$\boxed{\int (2x + 1)^3 e^{-x/2} dx = e^{-x/2} \left[-2 (2x + 1)^3 - 24 (2x + 1)^2 - 192 (2x + 1) - 768\right] + C}$$