What exactly is dx in an integral

[A.J.710]

I'm doing integration by parts and I am a bit confused as to what exactly dx is. Usually when integrating it is just dropped or forgotten about. Now when doing integration by parts there are some problems where you pick x as your u and dx as your du. since du is the derivative of u why isn't the du of x just 1 like we were always taught? Why is it now dx?

 

[Mark44]

I'm doing integration by parts and I am a bit confused as to what exactly dx is.

Mostly it indicates the variable with respect to which you're integrating. It also is suggestive of the Δx in a Riemann sum.

Usually when integrating it is just dropped or forgotten about.

That's not a good thing to do. dx doesn't play much of a role for very simple integrals such as substitutions. However, in integration by parts of trig substitutions, if you omit it, you will run into problems.

Now when doing integration by parts there are some problems where you pick x as your u and dx as your du. since du is the derivative of u

No, du is the differential of u. That's not the same as the derivative.

why isn't the du of x just 1 like we were always taught?

I doubt that's what you were taught. "du of x" makes no sense.
The derivative, with respect to x, of x is 1. In symbols, $\frac{d}{dx}x = 1$, but the differential of x is dx.

Why is it now dx?