1. **State the problem:** Find the integral of the function $2x^2$ with respect to $x$. 2. **Recall the formula:** The integral of $x^n$ with respect to $x$ is given by $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration. 3. **Apply the formula:** Here, the function is $2x^2$. We can factor out the constant 2: $$\int 2x^2 \, dx = 2 \int x^2 \, dx$$ 4. **Integrate $x^2$:** Using the formula, $$\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$$ 5. **Multiply by 2:** $$2 \times \frac{x^3}{3} + C = \frac{2x^3}{3} + C$$ 6. **Final answer:** $$\int 2x^2 \, dx = \frac{2x^3}{3} + C$$ This means the antiderivative of $2x^2$ is $\frac{2x^3}{3} + C$, where $C$ is any constant.