I am currently trying to learn about integrals, but I have run into a bit of an issue with the notation. In my calc book they give this definition for integration by parts...

I am able to understand why the following works and use it to do the same problems

Can someone explain why these equations are the same. If I am correct, when you examine the derivative

I know that you can define dx to be an independent variable so that

But how is this equation related to the dx in the integral. I don't seem to understand why the simplification is allowed. From what I have read, wikipedia says that the dx is related to how you define the integral (in terms of a theorem). I suppose I am using a Riemann Sum as a definition if that is what is meant.

I would simply ignore the notation and just do it my way, but in later chapters on differential equations they use differentials and integrals heavily, or so it looks like.

Thanks for any help, I have done quite a bit of work to no avail.