Assume you have the double integral,
$$ \iint f(x,y)dxdy $$ and want to find the answer by first integrating on $x$ and then integrating on $y$. we know that for each partial integration there is a constant. Now my question is, how to find the constant in the real applications?
I really appreciate if somebody explains that with an example.
The real question is why you would want to calculate an indefinite double integral. If for some really weird reason you wanted to, you would first write a constant for the integration by $x$, and when integrating with respect to $y$ you would integrate this constant and then include a new constant. Something like this is done when finding the potential function of a vector field.
In any "real application" you would not be calculating an indefinite integral (this is a terrible notation and worse name for an antiderivative), so this is a non-issue.