1. **Stating the problem:** We want to find the integral $$\int x^2 e^x \, dx$$. 2. **Method:** Use integration by parts, where we let: - $$u = x^2$$ so that $$du = 2x \, dx$$ - $$dv = e^x \, dx$$ so that $$v = e^x$$ 3. **Apply integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$ So, $$\int x^2 e^x \, dx = x^2 e^x - \int e^x (2x) \, dx = x^2 e^x - 2 \int x e^x \, dx$$ 4. **Integrate $$\int x e^x \, dx$$ using integration by parts again:** Let: - $$u = x$$ so $$du = dx$$ - $$dv = e^x \, dx$$ so $$v = e^x$$ Then, $$\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C$$ 5. **Substitute back:** $$\int x^2 e^x \, dx = x^2 e^x - 2 (x e^x - e^x) + C = x^2 e^x - 2 x e^x + 2 e^x + C$$ 6. **Factor the expression:** $$\int x^2 e^x \, dx = e^x (x^2 - 2x + 2) + C$$ **Final answer:** $$\boxed{\int x^2 e^x \, dx = e^x (x^2 - 2x + 2) + C}$$