Introduction

You already know how to differentiate a function — for instance, the derivative of $x^2$ with respect to $x$ is $2x$. Integration asks the reverse question: given a function, can we find another function whose derivative is the one we started with? If $\frac{d}{dx}F(x) = f(x)$, we call $F(x)$ an anti-derivative (or primitive, or integral) of $f(x)$.

Here's the subtlety: the anti-derivative is never unique. Since the derivative of any constant is zero, $x^2$, $x^2+5$, and $x^2-1$ all have the same derivative, $2x$. To capture every possible anti-derivative in one expression, we write $F(x)+C$, where $C$ is an arbitrary constant — this is why the result is called an indefinite integral.

In this notation, $f(x)$ is the integrand, $x$ is the variable of integration, and $C$ is the constant of integration. Because $C$ can take any value, the expression represents not one function but an entire family of functions — geometrically, an infinite family of curves, each obtained by sliding any one member of the family up or down along the $y$-axis. For example, the anti-derivatives of $2x$ trace out the family of parabolas $y=x^2+C$, all sharing the same shape but with vertices at different heights on the $y$-axis. …