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Why is there a differential in an indefinite integral?

  1. Mar 1, 2014 #1

    Jow

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    I understand why a definite integral of the form [itex]^{b}_{a}[/itex]∫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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  3. Mar 1, 2014 #2

    mathman

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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.
     
  4. Mar 1, 2014 #3

    PeroK

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    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!

    [tex]\int f(u(x))u'(x) Ix = \int f(u) Iu[/tex]
     
  5. Mar 1, 2014 #4
    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.
     
  6. Mar 1, 2014 #5

    Jow

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    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
     
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