1. The problem is to find the integral of the function $2x^2$ with respect to $x$. 2. The formula for integrating a power function $x^n$ is: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration. 3. Applying this to $2x^2$, we factor out the constant 2: $$\int 2x^2 \, dx = 2 \int x^2 \, dx$$ 4. Using the power rule for integration: $$2 \int x^2 \, dx = 2 \cdot \frac{x^{2+1}}{2+1} + C = 2 \cdot \frac{x^3}{3} + C$$ 5. Simplify the expression: $$2 \cdot \frac{x^3}{3} + C = \frac{2}{3} x^3 + C$$ 6. Therefore, the integral of $2x^2$ is: $$\boxed{\frac{2}{3} x^3 + C}$$