First Fundamental Theorem of Calculus

Let $f$ be a continuous function of a real variable defined on a closed interval $[a,b]$.
Let $F$ be the function defined for all $x\in [a,b]$ by

Then $F$ is uniformly continuous on $[a,b]$ and differentiable on the open interval $(a,b)$,
and $\frac{dF}{dx}(x)=f(x)$ for all $x\in (a,b)$,
so $F$ is an antiderivative of $f$.

Second Fundamental Theorem of Calculus

Let $f$ be a real-valued function on a closed interval $[a,b]$.
Let $F$ be a continuous function on $[a,b]$ which is the antiderivative of $f$ in $(a,b)$, i.e. $F^{\prime }(x)=f(x)$.
If $f$ is Riemann integrable on $[a,b]$, then