1. Let's start by stating the problem: understanding the integral rule in calculus. 2. The integral rule helps us find the antiderivative or the area under a curve of a function. 3. The most basic integral rule is the Power Rule for integration, which states: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1$$ 4. Here, $n$ is any real number except $-1$, and $C$ is the constant of integration because integration is the reverse of differentiation and can differ by a constant. 5. Important rules to remember: - The integral of a sum is the sum of the integrals: $$\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$$ - Constants can be factored out: $$\int a f(x) \, dx = a \int f(x) \, dx$$ 6. Let's do an example: find $$\int (3x^2 + 2x + 1) \, dx$$ 7. Apply the integral to each term: $$\int 3x^2 \, dx + \int 2x \, dx + \int 1 \, dx$$ 8. Use the power rule: $$3 \cdot \frac{x^{2+1}}{2+1} + 2 \cdot \frac{x^{1+1}}{1+1} + x + C = 3 \cdot \frac{x^3}{3} + 2 \cdot \frac{x^2}{2} + x + C$$ 9. Simplify: $$x^3 + x^2 + x + C$$ 10. So, the integral of $$3x^2 + 2x + 1$$ is $$x^3 + x^2 + x + C$$. This is how the integral rule works to find antiderivatives and areas under curves.