How do you actually use the word 'integral'? Recently I've noticed that I use the word "integral" to refer to two distinct concepts. I want to express myself more clearly, so my question is:

Question. Given a function $f:[a,b] \rightarrow \mathbb{R}$, are either of the following uses of the word 'integral' more standard and/or correct than the other? If so, what word or phrase should be used to refer to the other notion?
  
  
*
  
*Speaking of the integral of $f$, meaning the number $\int_a^b f(x) dx.$
  
*Speaking of an integral of $f$, meaning an $F:[a,b] \rightarrow \mathbb{R}$ with $F'=f$.

 A: This is the distinction between definite and indefinite integral.
When someone says integral, they can mean both definite and indefinite integral. It can also be the solution of a differential equation, so just like in many cases, it is the context which should give away what meaning has to be given to the word integral.
A: In Riemann integration of functions $\Bbb{R} \to \Bbb{R}$, there are three concepts about "integral" under the three names "definite integral", "indefinite integral", and "primitive". (For illustration ease I use "indefinite integral" in an old-fashion way.) 
The limit of the Riemann sum as the sequence of partitions "shrinks" defines the concept of definite integral. The limit of any function is unique; this justifies the "the" before "integral" in your first statement.
Given two functions, the first is called a primitive of the second if the derivative of the first is the second. The existence of a primitive of a continuous function $f: [a,b] \to \mathbb{R}$ turns out to be easily justified; a handy example is any function $F_{c}: x \mapsto \int_{c}^{x}f$ on $[a,b]$ where $c \in [a,b]$. Let us call any $F_{c}$ an indefinite integral of $f$; there being an "an" before "indefinite integral" is due to that $c$ is not unique. This existence result is called the fundamental theorem of calculus. Since $F_{c}$ is a primitive of $f$ for every $c \in [a,b]$, it follows that any constant shift $F_{c} + K$ is also a primitive of $f$. From here you can see the so-called second fundamental theorem of calculus.
As for the second statement, the author must use "an integral" to refer to "a primitive", regarding the requirement that $F'=f$.
