Why is there a differential in an indefinite integral?

[Jow]

I understand why a definite integral of the form $^{b}_{a}$∫ƒ(x)dx has the differential dx in it. What I don't understand, and what my teacher hasn't explained is why an indefinite integral (i.e. an antiderivative) requires the differential. Why does ∫ƒ(x)dx require that dx to mean "anti-differentiate"? To put it another way, why is the notation for the antiderivative an integral? It's obviously more than a question of notation, however, as without the differential techniques like u-substitution don't work. I hope this question makes sense, but this is something that has been bothering me for a while now. I figured my teacher would get to it eventually, but he hasn't yet.

 

[mathman]

Short answer. It tells you what is the variable for integration. Analog is d/dx to tell you the derivative is with respect to x.

 

[PeroK]

Why would techniques like substitution not work, or be dependent on notation? You could replace dx by "Ix" if you wanted and substitution would still work!

$$\int f(u(x))u'(x) Ix = \int f(u) Iu$$

 

[gopher_p]

To put it another way, why is the notation for the antiderivative an integral?

It's actually the other way around; we use the notation for integrals to talk about antiderivatives. The Fundamental Theorem of Calculus provides a link between the two concepts.

I am not aware of any notation specifically used to talk about antiderivatives, though I propose we begin using $\frac{d^{-1}}{dx}f$ to denote the family of functions whose derivatives are $f$ (which I'm certain will never happen). Most modern textbooks that I'm familiar with either don't have a notation until integration is introduced or they "cheat" and use the usual notation, $\int f\ dx$, before students learn about integration with little to no explanation for why that is the chosen notation.

In my opinion, referring to antiderivatives as "indefinite integrals" and denoting them by $\int f\ dx$ is an unnecessary abuse of terminology and notation that serves more to confuse students (and some calculus teachers, I'm sure) than it helps them communicate mathematical ideas.

 

[Jow]

Actually, I think I've resolved my confusion on the matter by searching around on the internet about differentials. The following link helped clear the issue up a bit: http://mathforum.org/library/drmath/view/65462.html