1. The problem is to understand how to compute an integral, which is the process of finding the area under a curve or the antiderivative of a function. 2. The basic formula for an indefinite integral is: $$\int f(x)\,dx = F(x) + C$$ where $F'(x) = f(x)$ and $C$ is the constant of integration. 3. Important rules include: - The integral of a sum is the sum of the integrals: $$\int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx$$ - The power rule: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$ 4. Example: Compute $$\int 3x^2\,dx$$ 5. Using the power rule: $$\int 3x^2\,dx = 3 \int x^2\,dx = 3 \cdot \frac{x^{2+1}}{2+1} + C = 3 \cdot \frac{x^3}{3} + C$$ 6. Simplify: $$3 \cdot \frac{x^3}{3} + C = \cancel{3} \cdot \frac{x^3}{\cancel{3}} + C = x^3 + C$$ 7. So the integral of $3x^2$ is: $$x^3 + C$$ This process can be applied to many functions to find their antiderivatives.