“Discovering” the indefinite integral's notation

Question

I'm currently reading Keisler's Elementary Calculus -- An Infinitesimal Approach, which develops the main results usually thought in undergrad calculus using Robinson's hyperreal numbers (instead of the more common "$\epsilon - \delta$ approach"). After the demonstration of the Fundamental Theorem of Calculus---which shows that the function $F(x) = \int_a^x f(t)dt$ is an antiderivative of the function $f$---it is just stated that the family of antiderivatives of the function $f$ will be denoted as $\int f(x)dx$. My question is, why does this notation work? In particular when doing integration by substitution, it is clear that the notation works "as expected" (lacking a better way to describe it).

To put the question another way, supposing I had just discovered the Fundamental Theorem of Calculus, how should I then reason in order to conclude that the notation $\int f(x)dx$ is a good one?

Answer 1

One path to discovering that notation on your own would be as follows:

If $a$ has no effect or can otherwise be ignored, then there is not much point in having notation that includes $a$. In these circumstances, you might abbreviate

$$ \int_a^x f(t) \, dt $$

to

$$ \int_{\bullet}^x f(t) \, dt $$

replacing $a$ with a mark saying something is there but you don't care what it is. You might then progress to just writing

$$ \int^x f(t) \, dt $$

Eventually, you realize that if you're just going to replace it with $x$ anyways, why bother introducing the new variable $t$? Then at this point you'll progress to

$$ \int f(x) \, dx $$

Then, when you reflect and decide you want a rigorous specification of what you mean, you invoke the usual trick of letting the notation mean all things it could have meant: i.e. it is the set of functions that come from making all choices of $a$ (or maybe a "variable" ranging over some set). And then you'd realize this is awkward, so you instead decide to include all anti-derivatives, not just the ones that come from definite integrals of the shape $\int_a^x$.

(you might make this last choice much earlier if you are frequently interested in the "$F$ is an antiderivative of $f$" problem)