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?

  • $\begingroup$ I'd have to read it to be sure, but my guess is that it's not a good notation at all because you'll be using the same symbol with two different meanings. One of the meanings is the set of all antiderivatives, the other is one antiderivative of $f$ (which an element of the set of all antiderivatives, by the way). $\endgroup$ – Git Gud Jan 13 '13 at 11:58

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

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


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

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)


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