1. The problem asks us to find the indefinite integral of the function $x^2$, which means finding a function $F(x)$ whose derivative is $x^2$. 2. Recall the power rule for integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ for any real number $n \neq -1$. 3. For the function $x^2$, $n=2$. Applying the power rule, we get: $$\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$$ 4. Here, $C$ is the constant of integration representing any constant value that could be added since differentiation eliminates constants. Final answer: $$\int x^2 \, dx = \frac{x^3}{3} + C$$