Anti-derivatives

Anti-derivatives are not integrals

If you have taken calculus in high school, you probably have seen the idea of an integral, and you may be wondering why we have been talking about anti-derivatives instead of integrals. The reason is that they're totally different concepts!

An anti-derivative is a function whose derivative is the original function.
As we'll see in the next section, an integral is the limit of a sum, computing a whole quantity as a sum of its pieces: $$\int_a^b f(x)\, dx = \lim_{N \to \infty}\, \sum_{i=1}^N \,f(x_i^*)\,\Delta x.$$

The reason that the two are frequently confused is because of an important theorem, called the Fundamental Theorem of Calculus, that relates derivatives and integrals. It says that:

You can get the antiderivatives of $f(x)$ from integrals of $f(x)$, and
You can compute integrals of $f(x)$ from an anti-derivative $F(x)$.

In fact, this theorem is so useful that the main way that we compute integrals is via anti-derivatives. But they're not the same.

You need anti-derivatives to compute integrals, just as you need the key to your front door to get into your house $\ldots$ but a key is not a house, and an anti-derivative is not an integral.