Would I be right to think that $$\int dx \,\,\,\frac{\partial}{\partial x} f(x,y)=f(x,y)$$ Or are there pathological cases?
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$\begingroup$If $x$ and $y$ are independent variables (and thus the $y$ is held constant during integration), then it is true that
$$ \int \frac{\partial f}{\partial x} dx = f(x,y) + C(y) $$ where $C(y)$ is equivalent to the integration constant for the univariate case. As such, up to the "constant", you are right.
If $y=y(x)$, then it is not that simple. For instance, if $f(x,y)=x^2-xy+y^2$ and you integrate along the line $y=2x$, then you are actually integrating $$ \int \frac{\partial f}{\partial x} dx = \int (2x-y) dx = \int 0 dx = 0 $$
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