1. **Problem Statement:** Find the antiderivative (indefinite integral) of the function $f(x) = 5x^2$. 2. **Formula Used:** The antiderivative of a power function $x^n$ is given by $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration and $n \neq -1$. 3. **Apply the formula:** Here, $n=2$, so $$\int 5x^2 \, dx = 5 \int x^2 \, dx = 5 \cdot \frac{x^{2+1}}{2+1} + C = 5 \cdot \frac{x^3}{3} + C$$ 4. **Simplify:** $$\frac{5}{3} x^3 + C$$ 5. **Explanation:** We increased the exponent by 1 (from 2 to 3) and divided by the new exponent. The constant 5 is factored out and multiplied after integration. The constant $C$ represents any constant term since the derivative of a constant is zero. **Final answer:** $$F(x) = \frac{5}{3} x^3 + C$$