Integration as Antidifferentiation

Differentiation takes a function to its rate of change. Integration reverses the process: given a rate of change, it recovers the original function. Because it undoes differentiation, integration is also called antidifferentiation, and it is the second pillar of calculus.

The indefinite integral. If $F'(x) = f(x)$, then $F$ is an antiderivative of $f$, and we write $$\int f(x)\,dx = F(x) + C.$$ The symbol $\int$ is an elongated "S" (for sum, as the next lesson explains), $f(x)$ is the integrand, and $dx$ indicates the variable of integration. The $+C$ is the constant of integration, and it is essential: differentiation destroys constants (the derivative of any constant is $0$), so when we reverse the process we cannot know what constant was there. Every antiderivative differs from every ot