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This question is concept-as-opposed-to-calculation based. I understand that when one sees the integral sign, followed by f(x)dx, that we can think of this as the indefinite integral, or antiderivative of f(x), with "respect to x," where the "dx" means just that ("with respect to x"). Or, we can think of the same expression as being in differential form, where "dx" no longer means "with respect to x," but now is interpreted as the "differential of x," a concept that becomes a useful tool when dealing with substitution techniques and the like. What I'm missing, in terms of understanding, is the following: Say that we integrate f(x)dx, where we think of "dx" as "with respect to x," and we get the function F(x) + C, where C is an arbitrary constant. The idea of the differential form, or at least my understanding of it, is that if we integrate f(x)dx, where "dx" is the "differential of x", we get the same antiderivative, F(x) (+ an arbitrary constant C). How is this so? If we think of "dx" as the "differential of x," is that not some quantity anywhere between (-infinity,infinity)? Are we not then multiplying f(x) by some quantity "dx" by this interpretation, as opposed to the interpretation that "dx" simply means

"with respect to x".

I understand how to manipulate the material to get the correct answer, such as using u-substitution, etc, but the concept does not sit well, so I am obviously missing something. Any help would be greatly appreciated.

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# Interpretation of dx as the differential of x for Indefinite Integrals

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