34

4

I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols.

For example, when one writes $$ \int \sin(x) \,\mathrm{d}x = -\cos(x) + k$$ then the status of both $x$ and $k$ is pretty unclear (which quantifier in front of each of these variables?)

Of course, I personally know how to translate this sequence of symbols into a proper mathematical sentence, but for students it seems utterly difficult to give a precise meaning to this, in particular at the stage when we try to explain the distinction between a function and its value at a point, or when we consider functions of several variables.

In my experience, this kind of notation tend to reinforce the student's habit to see mathematical notation as a kind of voodoo formulas that can be manipulated using certain incantations: no one probably knows what the incantation mean, but using the wrong incantation is forbidden for some reason (maybe it will summon an efreet?). On the contrary, I would like to show them the *meaning* behind everything we teach them.

For this reason, I try to never use indefinite integrals, relying instead on moving bounds, e.g.: $$ \forall a,x \quad \int_a^x \sin(t) \,\mathrm{d}t = -\cos(x)+\cos(a).$$

Questions:what possible issues are there in avoiding completely indefinite integrals? Is there any pedagogical advantage to using them? Is there a third way to go?

**Edit:** let me add another issue with the notation
$$ \int \sin(x) \,\mathrm{d}x = -\cos(x) + k$$
In the right-hand side, $x$ is implicitly a variable (as opposed to the *parameter* $k$), but on the left-hand side it is both a global variable and a local (mute) variable of integration. Given the (already somewhat weird) role we give to the integration variable in definite integrals, this is a source of confusion that bothers me a lot. Does anyone even imagine writing something like
$$ \sum_n n^3= \frac{n^2(n+1)^2}4+k?$$

5Do not omit indefinite integrals completely. After completing your class, the students will have to recognize and work with them. – Gerald Edgar – 2015-10-22T13:57:12.713

1Even if one takes the trouble to develop a self-consistent, coherent, optimized notational and conceptual system, there is no enforcement mechanism (well, ...) to make people behave sensibly in this or any other way. In particular, I guess we find ourselves needing to teach people how to cope with ambiguity or amorphousness, rather than telling them that there are absolutely-reliable universal conventions that will never change, etc. We even find ourselves forced (!) to deal with self-inconsistent, misleading, annoying conventions and "definitions". Dang. Hm. – paul garrett – 2016-04-14T23:23:03.363

8Just explain that it is a weird notation for the antiderivative, that the class might want to change, but you won't be able to go against the mathematical establishment – vonbrand – 2014-05-18T13:28:11.597

15Additionally, $+ C$ is not always sufficient. For example, the general antiderivative of $1/x$ is $$\begin{cases} \ln(x) + C_1 & \text{when $x > 0$,} \ \ln(-x) + C_2 & \text{when $x < 0$.} \end{cases}$$ – François G. Dorais – 2014-05-18T14:36:37.857