1. Let's solve an integral example: Find the integral of $f(x) = 3x^2$ with respect to $x$. 2. The formula for the integral of 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 formula to $3x^2$, we factor out the constant 3: $$\int 3x^2 \, dx = 3 \int x^2 \, dx$$ 4. Using the formula for $\int x^2 \, dx$: $$\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$$ 5. Substitute back: $$3 \times \frac{x^3}{3} + C = x^3 + C$$ 6. Therefore, the integral of $3x^2$ is: $$\int 3x^2 \, dx = x^3 + C$$ This means the antiderivative of $3x^2$ is $x^3$ plus a constant $C$.