I'm trying to make sure I have a good understanding of the fact that any two indefinite integrals of a function differ from each other by a constant value (specifically by a certain definite integral of the function).
Does the explanation I provide below display a correct understanding of the property?
Let $f$ be a function integrable over $[a,b]$, where $a\neq b$.
Let $n$ & $q$ be chosen fixed values in $[a,b]$ such that $n < q$.
Thus we can define two indefinite integrals of $f$, $N$ & $Q$, such that:
$$N(x)=\int_{n}^{x}f(t)dt\\\\Q(x)=\int_{q}^{x}f(t)dt$$
where $x$ is a variable value varying between $n$ & $b$ inclusive in the case of $N$ and varying between $q$ & $b$ inclusive in the case of $Q$.
Observe that, $\forall x\in[q,b]$,
$$N(x)-Q(x)=\int_{n}^{q}f(t)dt$$
In other words, for all $x$-values shared by the domain of $N$ & the domain of $Q$, the value of $N$ at $x$ differs from the value of $Q$ at that same $x$ by a fixed, constant amount. Namely, by $\int_{n}^{q}f(t)dt$.
Why does this property hold?
It follows from the definition of $N$, and the additive property of definite integrals, that for any $N(x)$ such that $x\in[q,b]$, we have:
$$N(x)=\int_{n}^{x}f(t)dt=\int_{n}^{q}f(t)dt+\int_{q}^{x}f(t)dt$$
Furthermore, it follows from the definition of $Q$ that, for any $Q(x)$ such that $x\in[q,b]$, we have:
$$Q(x)=\int_{q}^{x}f(t)dt$$
Thus we have
$$N(x)=\int_{n}^{q}f(t)dt+Q(x)$$
Rearranging, we find that
$$N(x)-Q(x)=\int_{n}^{q}f(t)dt$$
Thus, the property holds $\forall x\in[q,b]$.
Finally, notice that this property holds for any two indefinite integrals of $f$; all possible choices of $n,q\in[a,b]$ correspond to all possible pairs of indefinite integrals we can create $[a,b]$.
Best Answer
From comments, I learned that you are using Tom Apostol, Calculus I. I am guessing second edition (which is the edition I own).
As far as I could find, the term indefinite integral is introduced in section 2.18 and all examples of indefinite integrals have the form you use in your question. So although some other authors use the term indefinite integral differently, that is how Apostol uses it in Calculus I and your question should be read the way he uses it there.
(So my early comments about indefinite integrals -- where I had in mind that two indefinite integrals of $f$ might differ by more than the largest indefinite integral of $f$ -- might have been appropriate in some other context, but in this specific context, they were not.)
I that light I think what you have done is fine. You were very careful (maybe even more careful than Apostol in section 2.18).
Note that elsewhere you may see the term indefinite integral of $f$ used differently than in Apostol's book. It may refer to the family of functions containing all primitives of $f$ (Apostol defines primitives in section 5.3) or it may be used interchangeably with antiderivative (which Apostol uses interchangeably with primitive). This is one reason why knowing your definitions -- and knowing which definitions apply in a particular context -- is important! I raise the points in these paragraphs not with the intent to confuse you, but in an attempt to avoid having you encounter these slightly different definitions elsewhere without having been warned that such a thing might happen.
Collecting some thoughts from comments of mine that probably should have been answers:
I did not find a formal definition of indefinite integral anywhere in Apostol, but there is a statement that an expression such as $\int_a^x f(x)\,dx$ for constant $a$ is "sometimes" called an indefinite integral of $f$, and the idea seems to be that only such expressions are called indefinite integrals. Later, Apostol formally defines an antiderivative of $f$ to be a function whose derivative is $f$ (that is, $\int f(x)\,dx$ for a particular choice of integration constant). And you are right, that can only occur later because it requires the definition of a derivative.