Does the notation $\int f(x,y)\partial x$ make sense $\left(\text{where } f(x,y)=\dfrac{\partial g(x,y)}{\partial x}\right)$?

Question

I am currently studying PDE's where the integration of partial derivative is commonplace. I have seen that in a lot of literature the integral of a partial derivative is just written as

$$\int f(x,y)dx$$

But I find that if you were to write something like $\int f(x,y)\partial x$ it would be much clearer that the integral will have an arbitrary function rather than just a constant. In my head this makes sense but I was wondering if there reason this convention is not widely adopted or if it is even correct to write an integral like this at all?

Answer 1

You are not integrating a derivative or a partial derivative of $f$, but just the function $f$ itself with respect to one variable. You can see $\mathrm dx$ (the Lebesgue measure) as the derivative of the function $x$, but then it only depends on one variable, so there is no reason to write a $∂$ symbol. See also here for more about the notation $\mathrm dx$.

The notation $\mathrm d f$ on the contrary was already chosen to be the (total) differential of $f$, which is why one then has to denote with a $\partial f$ the partial derivatives.