TITLE: Integrating a Partial Derivative
QUESTION [25 upvotes]: Would I be right to think that $$\int dx \,\,\,\frac{\partial}{\partial x} f(x,y)=f(x,y)$$
Or are there pathological cases?

REPLY [32 votes]: 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
$$
\int \frac{\partial f}{\partial x} dx = \int (2x-y) dx = \int 0 dx = 0
$$