How useful/useless is the indefinite integral

Question

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 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 for differentiation. But still, as a function it is not completely unambiguous. Of course, there are applications where this additive uncertainty (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.

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 MESE: (Link)

Answer 1

The way I teach it, $\int f(x)dx$ is a symbol which stands for the entire set of solutions to the differential equation $\frac{dy}{dx} = f(x)$. The content of the fundamental theorem is that to evaluate a definite integral on an interval, one need only evaluate the difference between the value of the end points for any member of this set.

I do agree that the notation is confusing, because it looks like it represents one function, when in fact it stands for a whole class of functions. Even writing $+C$ does not fix everything, since your domain might be disconnected. Well, it does fix everything if you interpret $C$ as a locally constant function...

Answer 2

Is indefinite integration overrated? Yes, in my opinion.

Is indefinite integration necessary? Yes, in my opinion.

You can't be a mathematician (or a physicist, or an engineer) if you can't compute antiderivatives, and therefore we should teach indefinite integration. We can write $I(f)$ instead of $\int f(x)\, dx$, or any other symbol we like, but our students should learn to compute antiderivatives.

Answer 3

I will echo Steven Gubkin's comment and say that it is important for calculus students to be able to recognize the notation $\int f(x) dx$ and understand what it means, because of the existing convention of using it.

However, I can also make the observation that in intro calculus classes, at least as far as I know, the (admittedly subtle) distinction between an antiderivative and an indefinite integral is usually glossed over. So I don't think much would be lost at that level by, say, avoiding the phrase "indefinite integral" and using "antiderivative" instead.

Either way, you have a point that, at least in applications of mathematics, integrals tend to be definite. In physics it is very common to write $\int f(x) dx$ to mean a definite integral where the limits are left implicit, either because they are clear from context or because they're irrelevant. (This extends to things like $\int^b f(x) dx$ which I see from time to time as well.)

Answer 4

Given a continuous function $f:\>I\to{\mathbb R}$ on some interval $I\subset{\mathbb R}$ the indefinite integral $$\int f(x)\>dx\tag{1}$$is not a function but a set of functions, namely the set of all functions $F:\>I\to{\mathbb R}$ satisfying $F'(x)=f(x)$ for all $x\in I$. Such functions $F$ are called primitives of $f$, and one proves that they are all equal up to an additive constant.

One wants to be able to talk about the primitives of a given $f$ even before "computing" a single one of them, and the notation $\int f(x)\>dx$ for the full set comes as a handy notation.

It so happens that we often want to talk about a difference $F(b)-F(a)$, where $F$ is a primitive of $f$ which we are not jet in possession of. Such a difference can then be denoted by $$\int_a^b f(x)\>dx\ ,\tag{2}$$ which is called the definite integral of $f$ from $a$ to $b$. Even though there are an infinity of primitives of $f$ the difference $F(b)-F(a)$ has the same value for all of them; so it is allowed to denote it by the term $(2)$ that doesn't exhibit the $F$ at all. Note that $(2)$ is defined whether $a<b$ or $a>b$.

All this is more or less formal and quite trivial. It only has to do with taking derivatives and the inverse of this process. The fundamental theorem of calculus is a totally different matter.