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.