Antiderivatives of Elementary Functions
The Big Idea
If differentiation answers "What is the rate of change?", then antidifferentiation answers the reverse question: "What function has this as its derivative?" Finding antiderivatives is the foundation of integral calculus.
Formal Definition
A function $F(x)$ is called an antiderivative of $f(x)$ on an interval $I$ if:
$$F'(x) = f(x) \quad \text{for all } x \in I$$
The general antiderivative (or indefinite integral) of $f(x)$ is written as:
$$\int f(x)\, dx = F(x) + C$$
where $C$ is an arbitrary constant called the constant of integration.
Why the $+ C$? If $F(x)$ is an antiderivative of $f(x)$, then so is $F(x) + C$ for any constant $C$, because the derivative of a constant is zero. For example, $x^2$, $x^2 + 5$, and $x^2 - 17$ are all antiderivatives of $2x$.