Let:
$\displaystyle F(x,y) = \int \int \frac{1}{xy} \, dx \, dy$
be a bidimensional indefinite integral. I would like to understand the correct way to handle it.
My first attempt would be to simply solve the integral inside in $dx$ and then the outer one in $dy$.
That is:
$\displaystyle F(x,y) = \int \frac{\log x}{y} + c_1 \, dy = \log(x) \log(y) + y \, c_1 + c_2$
In this way however I get a non symmetric result, starting from a symmetric argument.
The second attempt would be:
$\displaystyle F(x,y) = \int \frac{\log x + c_1}{y} \, dy = (\log x + c_1)(\log y + c_2)$
In this way I get a symmetric result. However I'm not sure it's correct to divide $c_1$ by $y$. In a way, $y$ is a constant if seen from the inside integral. However if I consider it constant, then why $y$ and not $y^2$.
Technically both solutions, if derived respect to $x$ and $y$ give the same starting argument.
Can you explain which one is correct (if one is) and why?