Examples. An indefinite integral (or antiderivative) of $\cos$ is $\sin$:
$$\int \cos = \sin.$$
Edit: There has been much unexpected confusion with the above statement. I define the above statement to mean precisely that an antiderivative of the cosine function (which has domain $\mathbb R$) is the sine function (which has domain $\mathbb R$). Or equivalently, the derivative of the sine function is the cosine function. This notation though is not what the question is about, so I hope you will not dwell on this. If you want to question the merits of this notation, that is perhaps for another question. If it really bothers you, you can just pretend the above is statement instead $\int \cos x = \sin x$ or $\int \cos x \, \textrm dx = \sin x$ or $\int \cos x \, \textrm dx = \sin x + C$. (I personally assign to these latter statements slightly different meanings. And yes, I do explain all of this very clearly and repeatedly to my students.)
The definite integral (or "area function") of $\cos$ from $0$ to $\pi/2$ is $1$:
$$\int^{\pi/2}_0 \cos = 1.$$
These are distinct concepts. But by custom, we use the same symbol for them, because the Fundamental Theorem of Calculus tells us that
$$\int^{\pi/2}_0 \cos = \left(\int \cos\right)\left(\frac{\pi}{2}\right)-\left(\int \cos\right)(0) = \sin \frac{\pi}{2}-\sin 0 = 1- 0=1.$$
This often causes students a great deal of confusion (see e.g. below "Related (but distinct) discussions"). In particular, students
- don't see any difference between the indefinite and definite integrals;
- believe that integration is, by definition, the inverse of differentiation; and
- thus fail to understand that the significance of the FTC or see that it is more than a mere definition.
To reduce the above confusion, I have already tried to follow @PeteL.Clark's recommendation (in this answer) of using the term antiderivative (rather than indefinite integral) as much as possible.
I am now thinking of also trying the following approach:
- First introduce antidifferentiation. Never once mention the word integral. Don't use $\int$ as our symbol for the antidifferentiation operator---instead, use some other symbol like $\square$. In which case, we'd write things like $\square \cos=\sin$.
- Introduce integration (i.e. the definite integral) as usual.
- Go through the FTC, explaining that integration turns out to be, in a certain precise sense, the "same thing" as antidifferentiation.
- Explain that because of the FTC (and custom), we'll now discard our $\square$ symbol for the antidifferentiation operator and replace it with $\int$.
I don't think I've ever seen any writer using this approach (but please let me know if you have). So, what disadvantages might the above approach have?
Pete L. Clark also writes
At some point I slip up because after all we are using the same integral sign for both: I almost wish we didn't. (Using almost identical notation for two a priori incredibly different things which in the setting of the FTC become almost the same is one of the brilliant notational innovations of calculus.
I don't quite understand the above quote. I understand that today, after centuries of use, it would be difficult and foolish to go against tradition and attempt to completely replace the $\int$ symbol for the indefinite integral/antiderivative with something else altogether.
But why wouldn't it have been better if mathematicians had just decided to use a different symbol from the start? What is so brilliant about using the same symbol $\int$ for "two a priori incredibly different things"?
It seems that this has simply resulted in a great deal of confusion (at least among average students of calculus), without any obvious gain in convenience or understanding.
Related (but distinct) discussions: