if f = f(x,y)

and ∂f/∂x = g(x,y)

does

f = ∫ g(x,y) ∂x

or

f = ∫ g(x,y) dx ?

When computing this integral do you consider y as a constant?

Thanks

use , though i think we only use that in partial differential equation courses (in my experience).

yes

Thanks .

I'm getting confused because I have a text book that says:

A(x,y) = ∂U(x,y)/∂x

And then states that from this

U(x,y) = ∫ A(x,y) dx + F(y)

I can see the reasoning for the F(y) function: if you partially differentiate A(x,y) you'll lose all exclusively y terms. But as you can see the differential is dx, not ∂x. Rather confusing! Any explanation? Thanks again.

Notation isn't perfect and often isn't standard. Don't lose any sleep over it. Again, often, the context must give you the necessary clues.

I would quarrel with the use of the term integral.
What you are really asking about is antiderivatives or in this case partial antiderivatives.
So in the case of the indefinite “integral” the or the simply indicates the variable with which we are working.
As you noted, the f(y) is constant when working with .