Integrating x and xdx: What's the Difference?

[Make]

Hello. I'm sort of confused about when you can integrate... Say for example if you have dx you're sort of integrating a "hidden" constant, right? So you increase the power by one and either use the upper and lower limit or just add c: x+c. And similarly if you have xdx, again you increase the power by one and divide:0.5x^2, right? But when are you allowed to integrate? Namely, must you always first obtain dx or dvariable somehow? How do I obtain this in the following case for example: Y=2x? If it is not necessary, then what difference is there in integrating x or xdx? THANK YOU SO MUCH FOR YOUR HELP!

 

[tiny-tim]

Hi Make!

I'm not sure what you're asking, but does this help? …

When you integrate a function f(x), you have to say what you're integrating it with respect to.

Usually, obviously, it's with respect to x, in which case we write either

"integral of f(x) with respect to x"

or

"∫ f(x) dx".

You can't write an ∫ without a d(something).​

As to your y = 2x example, you can write ∫ 2x dx, or ∫ y dx … they're the same (but they're not the same as ∫ y dy, because that's integrating with respect to something different).

 

[raymo39]

dead on answer for makes level of understanding :)