How to view 'dx' in an Integral?
The $dx$ is not an important part of the actual calculation, no. If you like the view that the integral notation means "integrate whatever is between $\int$ and $dx$", you can do that. However, it will make anything more advanced than simple anti-differentiation difficult to remember and do correctly.
Historically, $dx$ comes from the transformation that underlies the definition of integrals $$ \sum_i f(x_i)\Delta x \to \int f(x)dx $$ and it can be seen simply as a relic from that, if you want.
However, an actual meaning may be given to $dx$, and having one in mind that will make more advanced things like theoretical arguments and substitution a lot easier to do correctly.
In fact, there are several possible meanings, depending on your taste. I think the simplest one to explain intuitively may be the one from measure theory: $f(x)dx$ can here be seen as a density, and you integrate up density along a bit of the number line to find total mass. This would make $dx$ the standard, unit density of the number line, while $f(x)$ tells you the actual density at any point, in units of $dx$.
In the early 1970's, (I want to say 1973, but I'm not 100% certain) this question was debated in the letter column of several issues of the American Math Monthly. There were three schools of thought, as I remember.
- It's just a punctuation mark that just tells you where the integrand ends.
- It's a dummy variable.
- It's an essential part of the notation; we integrate differential forms, not functions.
No conclusion was reached, and the editors eventually refused to publish any more letters on the subject.
As for me, I'm not happy with any of these answers, but I have nothing to offer in their place.
The following was originally my comment, but I want to put it in an answer because a conclusion has been reached, which is contrary to the answer most above.
Go here $\longrightarrow$ https://m.youtube.com/watch?v=U_q7R5JJvb4 for a meaning of $\text{d}x$ in a general integral. In a Riemann Integral, the definition of $\text{d}x$ is a little bit different as it is derived from the Riemann Sum. For that, I suggest going here $\longrightarrow$ https://m.youtube.com/watch?v=Stbc1E5t5E4.
There have been definitions like treat $\text{d}x$ as a full stop but this is only to help notating an integral in symbols. This is not the definition of $\text{d}x$.
I also found another link to a YouTube video that is part of a series called Essence of Calculus. In this video, it specifically talks about integrals $\longrightarrow$ https://m.youtube.com/watch?v=rfG8ce4nNh0 however, this video is quite advanced and perhaps difficult to understand at first for there may be much to keep up with.