Can You Integrate a Function f(x,y) with Respect to y and Treat x as a Constant?

[Je m'appelle]

I just need to understand the following

\int f(x,y) \partial y\ is\ solvable\ for\ 'y'\ as\ the\ variable\ and\ 'x'\ as\ a\ constant\ while,

\int f(x,y) dy\ is\ impossible\ to\ solve.

Is this correct? If so, could anyone please provide me with evidence for this?

Thanks in advance.

 

[Mark44]

I just need to understand the following

\int f(x,y) \partial y\ is\ solvable\ for\ 'y'\ as\ the\ variable\ and\ 'x'\ as\ a\ constant\ while,

I understand what you're trying to say, but I don't recall ever seeing any integrals with the "partial y" symbol. The whole idea here is that you have some function involving x and y that you want to integrate with respect to y (or x). Even though the function has two variables, you treat one of them as a constant.

\int f(x,y) dy\ is\ impossible\ to\ solve.

This is actually done all the time when double or triple integrals are rewritten as iterated integrals. If a double integral is written as an iterated integral , the integrand is a function of two variables, and integration is performed first with respect to one of the variables, and then later, with respect to the other. When you integrate with respect to one variable, you are treating the other variable as if it were a constant.

Is this correct? If so, could anyone please provide me with evidence for this?

Thanks in advance.