Rules for Finding Antiderivatives
Big Idea: If differentiation tells us the rate of change, then antidifferentiation (also called indefinite integration) reverses the process — it recovers the original function from its derivative.
1. What Is an Antiderivative?
Definition: A function $F(x)$ is an antiderivative of $f(x)$ on an interval $I$ if
$$F'(x) = f(x) \quad \text{for all } x \in I$$
Because the derivative of a constant is zero, if $F(x)$ is one antiderivative of $f(x)$, then $F(x) + C$ (where $C$ is any real constant) is also an antiderivative. We write:
$$\int f(x)\,dx = F(x) + C$$
The symbol $\int$ is the integral sign, $f(x)$ is the integrand, $dx$ indicates the variable of integration, and $C$ is the constant of integration.