The other part of the Fundamental Theorem (sometimes called the first Fundamental Theorem) relates indefinite integrals to anti-derivatives:
| (First) Fundamental Theorem of Calculus:
If $f$ is a continuous function, then $$\frac{d}{dx} \int_a^x f(s)\, ds = f(x).$$ |
That is, the indefinite integral is an anti-derivative. The derivative of the (indefinite) integral is the original function.
Warning: The notation $\int f(x) \,dx$, without any upper and lower limits on the
integral sign, is used in two different ways. Sometimes it is used to mean
the indefinite integral $\int_a^x f(s) \,ds$, and
sometimes it is used to mean "an anti-derivative of $f(x)$". Since
$\int_a^x f(s)\, ds$ is an anti-derivative, this is consistent.