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I've come across these two notations for calculating an indefinite integral but I'm not sure whether or not they are equal:

  • $f(x)dx$
  • $\int f(x)dx$

When calculating the indefinite integral, the first notation is used in my learning book, but isn't that the same as the second notation?

Thanks for any clarification.

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Could you please provide a reference where the notation appears? The context might make it easier for people to explain the role of the notation. –  Jonas Meyer Apr 27 '11 at 18:18
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...the first is just a differential. –  J. M. Apr 27 '11 at 18:18
    
As J.M. says, if $\int f(x)dx = F(x)+C$, then $dF=f(x)dx$ is the differential. At least, this is the usual notation. –  Jonas Meyer Apr 27 '11 at 18:19
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2 Answers 2

up vote 4 down vote accepted

The first is not an integral; it's a differential. The second is an integral.

When doing substitution or integration by parts, one considers differentials. For example, to do integration by parts on $$\int x\sin x\,dx$$ one can say "let $u=x$ and $dv = \sin x\,dx$." Then you want to find a function whose differential is $dv$, so you are trying to find $\int dv = \int \sin x\,dx$; we usually don't actually write this, and simply write "then $v=-\cos x$", which may be what is confusing you and leading you to believe that "$\sin x\,dx$" is some kind of integral. But it's not an integral.

I'm just guessing, mind you, since you have not yet provided explicit examples of the use you believe refers to integrals.

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This makes sense. It looks like I was overlooking something. Thanks. –  pimvdb Apr 27 '11 at 18:27
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The first one usually denotes a differential 1-form, where the second denotes the indefinite integral of $f$, i.e. it's anti-derivative.

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