Why is there a differential in an indefinite integral?

Jow
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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.
 
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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.
 
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
 
Jow said:
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.
 

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