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I am confused about the constant in the double integral.

Let the integral be,

$$ F(x,y)+C = \int_y\int_x f(x,y) \, dx \, dy $$

Assume that the integral can be solved by firstly integrating on $x$ and then finding the resulting integral over $y$. Then my question is about having two constants in each step or just one in the very end?

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Rose
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  • The question of what an indefinite double integral ought to be is subtle. If you do the indefinite integration with respect to $x$, you get a constant... but actually you don't; what you get is an unknown function of $y$; that is, it's a constant for a fixed value of $y$, but for different values of $y$ it might be different constants. If you then do a second indefinite integral, you have a term for the integral of this unknown function, but you also introduce an unknown function of $x$. – Qiaochu Yuan Oct 17 '16 at 20:59
  • @QiaochuYuan Understand, but how in reality we find the constant? – Rose Oct 17 '16 at 21:13
  • The point is, it's not a constant at all, necessarily. Presumably you want to find all $F(x,y)$ such that $\frac\partial{\partial x}\frac\partial{\partial y}F(x,y) = f(x,y)$. In one variable, the family of all such antiderivatives is just one particular antiderivative plus an arbitrary constant. But here, there are many more antiderivatives. For example, in the simplest case $f(x,y)=0$, any function of the form $F(x,y) = g(x) + h(y)$ is an antiderivative. Can you show that these are all the antiderivatives? – Greg Martin Oct 17 '16 at 21:34

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