The definition of the Riemann integral is:
Let $f(x)$ be defined on a segment [a,b]. If a real number $I$ exists such that: $(\forall \epsilon > 0)(\exists b(\epsilon)>0)(\forall d)(||d||<b(\epsilon)) \rightarrow |S(f(x),d,a,b) - I| < \epsilon$ than $I$ is the Riemann integral of a function on a segment [a,b].
The question is:
What is the correct theorem that connects the Riemann integral of a function to it's primitive function?
I have two possible candidates. The first is:
Let $f(x)$ be continuous on an interval $I$. Then $f(x)$ has a primitive function on an interval $I$ and it's of the form $G(x)=\int_{a}^{x} f(t)dt$ where $a\in I$.
The second one is the Newton-Leibniz formula:
If $f(x)$ is continuous on an interval $I$ and if $F(x)$ is it's primitive function on that interval than for every segment $[a,b]\subseteq I$, $\int_{a}^{b} f(x)dx = F(b)-F(a)$.
Which of these satisfies the question if any one of them? If one does why does it satisfy over the other? They seem to use both definitions to make crucial results so which is correct?
Both. To justify this, we first note that the second proposition does not say anything about the existence of a primitive of $f$. It only assumes that a primitive of $f$ exists. Now to answer how to construct a primitive of a given function (under suitable conditions, of course) one utilizes the first proposition: The function $x \mapsto \int_{a}^{x}f$ is a primitive. The combination of these two theorems then give the desired connection.