In my book, "Elementary Differential Equations and Boundary Value Problems" $11$th ed. Ch $2.2$ Example $2$ the author integrates both sides of an equation with respect to $x$ and $y$. To my knowledge if you do something to one side of an equation you have to do the same to the other side of the equation so that the equation still holds. In the example the author integrates one side of the equation with respect to $x$ while integrating other side with respect to $y$. I could not make sense of it and I couldn't find an online integral calculator that integrates an equation for verification. Can you explain me what is going on?
My calculations for the Eq$(18)$ is $~y - 2 = x^2 + 2x + 2 + c~$ because integrating $y$ terms wrt $x$ and $x$ terms wrt $y$ is going to bring one $x$ and one $y$ multipliers to the equation.
I hope you understood what I mean. I am waiting for your answers. Thank you!