I have a question for all the mathematically inclined readers out there. You guys have probably already noticed that I’m a bit picky when it comes to mathematical notation, whereby “picky” I mean “anal-retentive to an alarming degree.” I’ve gone on about renaming the imaginary number or being precise with trig, and I’m a believer that was the wrong choice. In keeping with that theme, I have a notational question. It should be a basic one, but I found it was surprisingly more nebulous than I originally thought. Here it is:

What,

precisely, is an “indefinite integral”?That is, what does the symbol mean?

Lest you think I am (more of) a complete idiot, let me be more specific:

- Does this symbol refer to a single function?
- If not, does it refer to a family of functions?
- If so, what family?

Once you’ve made up your mind, then ask yourself this second question: which of the following responses

- (a)
- (b)
- (c)

is the correct answer to the question

I’ll give my answer to the first question after the jump, and explain how I’ve always been wrong about it.

Okay, on to the big question:

What,

precisely, is an “indefinite integral”?That is, what does the symbol mean?

I’ve always taken to refer to the family of all antiderivatives of . In particular, an *indefinite integral* is really a *set* of functions. More specifically,

,

which (assuming it’s nonempty!) is always an *infinite* family of functions, since whenever is an element of $latexint f(x) , dx$, then is as well.

If is defined on some interval , then the Mean Value Theorem implies that these are in fact the only possible antiderivatives; that is, if , then . As a result, if is any function defined on that same interval with , then

.

Of course, in the calculus classroom, we usually abuse the notation and drop the curly braces and just write

.

Hence, when I write something like

.

I’ve always taken it to mean that every antiderivative of is of the form for some appropriate value of .

(As an aside, I typically state computational bit of the Fundamental Theorem of Calculus as follows:

If is continuous on the interval , then

Of course, under my standard interpretation of the indefinite integral, this is another abuse of notation, for the definite integral on the left is a *number *whereas the expression on the right is, *a priori*, a set of *infinitely many* numbers. However, by virtue of the Mean Value Theorem, it is in fact a singleton set consisting of a unique number, which is, to me, part of the magic of the FTC.)

Okay, so far, so good.

Now, what about ? Every calculus book on the planet writes

.

Unfortunately, under the interpretation of an indefinite integral above, *this formula is wrong*! More precisely,

.

Certainly every function of the form is satisfies . However, there are other functions with that are *not* of this form. In particular, any function of the form

with constants also satisfies on the same domain as . However, while looks like for positive , it looks like for negative , so it is *not *a vertical translate of .

With the Mean Value Theorem, it is easy to show that *every *function with does take the form above, so according to my understanding of the indefinite integral, we should write

…but we don’t.

Clearly I’m *wrong* about something here… but where? What gives? What does an indefinite integral *really *mean?

I’ll share what I learned about this with you next week, but I’ll give you some time to resolve this quagmire for yourself first.

## Updates!

(1) Apparently when I updated WordPress, it changed the comment settings, and was only allowing “registered WordPress” users leave comments. Thanks to the folks who sent me email about this problem… hopefully it’s fixed now!