1. The problem is to find the integral of a function, which means finding the antiderivative or the area under the curve of the function. 2. The general formula for the indefinite integral is $$\int f(x)\,dx = F(x) + C$$ where $F'(x) = f(x)$ and $C$ is the constant of integration. 3. For example, to integrate $f(x) = x^2$, we use the power rule for integration: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$. 4. Applying this to $x^2$, we get $$\int x^2\,dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$$. 5. This means the antiderivative of $x^2$ is $\frac{x^3}{3} + C$, which represents all functions whose derivative is $x^2$. 6. Remember, the constant $C$ is important because differentiation of a constant is zero, so it accounts for all possible vertical shifts of the antiderivative.