# How useful/useless is the indefinite integral [duplicate]

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community.

Do you think it makes sense to teach indefinite integrals? My opinion is that only definite integration should be taught since it is the only one that makes formal sense to me. Of course indefinite integrals can be used by people who know what they are doing, but it doesn't justify the introduction of this notion from the very beginning to the layest of the laymen.

I would like to argue as follows:

1. One often read/hears $\int..dx$ is the inverse of differentiation, its the anti-derivative. While one can of course make some sense of it, of course everybody knows that differentiation is an irreversible operation where information is lost, so there is no true inverse of that operation. For me the usage of "anti-" in the sense of "almost-anti-" is one source of confusion.

2. In my opinion $\int f(x) dx$ should not be seen as a function, written like that, for my taste, I would say that it's not well-defined as a function. If it is a function, of what variable? Certainly not of $x$. It would make slightly more sense to write $\int^t f(x)dx$ as now at least one can use this form differentiation. But still, as a function it is not completely unambiguous. Of course, there are applications where this additive unsertainty (which can be infinity) does not play a role, but again this is of no concern for people who are just being taught what integrals are. I

3. The only sensible use of writing $\int f(x)dx$ that I can imagine is as a sort of abbreviation in the sense "you know what boundaries you have to insert, so let's just skip it". It is like writing sums without giving the boundaries: $\sum f(n)$, which I would generally avoid to do, unless everyone knows what is meant.

Given that I see school text books full of indefinite integrals from the beginning and that the search on math.stackexchanges for "indefinite integral" gives >1000 results, where sometimes calculations of this sort (Link) are carried out with the result that $\int\frac{dx}{x}=\ln(x)$ without anyone complaining about the notation which is at most sketchy, and finally that searching wikipedia for "indefinite integral" automatically redirects to "anti-derivative", I would like to ask, what do you think about using indefinite integrals in mathematics? Should school children be exposed to it? Should it be taught?

P.S.: this question has also been posted on SE: (Link)

• If we work on the set of equivalence classes of functions where two functions are equivalent if they differ by a constant then the indefinite integral and derivative are indeed inverse operations. See discussion around page 180 of supermath.info/LinearNotes.pdf – James S. Cook Jul 26 '14 at 17:51
• @JamesS.Cook If they differ by a locally constant function, you mean. If the domain is disconnected, then you need a different constant for each connected component. – Steven Gubkin Jul 26 '14 at 19:11
• @StevenGubkin but of course. I think I treated polynomials in the linear notes so... but, yes, disconnected domains always make things a bit more complicated. – James S. Cook Jul 26 '14 at 22:38