1. **State the problem:** We need to find the integral of the function $x^2 e^{2x}$ with respect to $x$. 2. **Formula and method:** We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ 3. **Choose parts:** Let $$u = x^2 \quad \Rightarrow \quad du = 2x \, dx$$ $$dv = e^{2x} dx \quad \Rightarrow \quad v = \frac{e^{2x}}{2}$$ 4. **Apply integration by parts:** $$\int x^2 e^{2x} dx = x^2 \cdot \frac{e^{2x}}{2} - \int \frac{e^{2x}}{2} \cdot 2x \, dx = \frac{x^2 e^{2x}}{2} - \int x e^{2x} dx$$ 5. **Integrate $\int x e^{2x} dx$ using integration by parts again:** Let $$u = x \quad \Rightarrow \quad du = dx$$ $$dv = e^{2x} dx \quad \Rightarrow \quad v = \frac{e^{2x}}{2}$$ Then, $$\int x e^{2x} dx = x \cdot \frac{e^{2x}}{2} - \int \frac{e^{2x}}{2} dx = \frac{x e^{2x}}{2} - \frac{1}{2} \int e^{2x} dx$$ 6. **Integrate $\int e^{2x} dx$:** $$\int e^{2x} dx = \frac{e^{2x}}{2} + C$$ 7. **Substitute back:** $$\int x e^{2x} dx = \frac{x e^{2x}}{2} - \frac{1}{2} \cdot \frac{e^{2x}}{2} = \frac{x e^{2x}}{2} - \frac{e^{2x}}{4}$$ 8. **Final substitution:** $$\int x^2 e^{2x} dx = \frac{x^2 e^{2x}}{2} - \left( \frac{x e^{2x}}{2} - \frac{e^{2x}}{4} \right) + C = \frac{x^2 e^{2x}}{2} - \frac{x e^{2x}}{2} + \frac{e^{2x}}{4} + C$$ **Answer:** $$\int x^2 e^{2x} dx = e^{2x} \left( \frac{x^2}{2} - \frac{x}{2} + \frac{1}{4} \right) + C$$