Fundamental theorem of calculus

This statement links the Riemann integral with differentiation.

Part I (integral defines an antiderivative where $f$ is continuous)

Let $f : [a, b] \to R$ be Riemann integrable and define

φ (x) := \int_{a}^{x} f (t) d t .

Then:

$φ$ is continuous on $[a, b]$ .
If $f$ is continuous at $x_{0} \in (a, b)$ , then $φ$ is differentiable at $x_{0}$ and

φ^{'} (x_{0}) = f (x_{0}) .

Part II (integral of a derivative)

If $f$ is a derivative on $[a, b]$ , i.e. there exists $F$ with $F^{'} (x) = f (x)$ on $[a, b]$ , then

\int_{a}^{b} f (x) d x = F (b) - F (a) .

A common special case (“Barrow’s theorem”): if $F$ is differentiable and $F^{'}$ is continuous on $[a, b]$ , then the same formula holds.