1. The problem asks to solve all given problems using integrals, but since no specific problems are provided, I will demonstrate how to solve a basic integral problem. 2. Consider the problem: Find the integral of the function $f(x) = x^2$. 3. The formula for the integral of a power function is: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$ and $C$ is the constant of integration. 4. Applying this formula to $f(x) = x^2$, we have $n=2$. 5. So, $$\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$$ 6. This means the antiderivative of $x^2$ is $\frac{x^3}{3} + C$. 7. This process can be applied to other polynomial functions similarly. Since no other problems were specified, this completes the solution for the first problem involving integrals.