I have been asked to find the "Most general solution" for $u(x,y)$ of the PDE
$$\frac{\partial u}{\partial x} = \frac{x}{\sqrt{x^2+y^2}} + 3y\cos(3xy) + 3x^2y^2$$
I know you must take the integral of both sides to "undo" the partial derivative. Is it reasonable to then split the right hand side into three separate integrals as below?
$$u(x, y) = \int\frac{x}{\sqrt{x^2+y^2}}dx + \int3y\cos(3xy)dx + \int3x^2y^2dx.$$
From here I got: $$\int\frac{x}{\sqrt{x^2+y^2}}dx = \frac{1}{2}\int\frac{2x}{\sqrt{x^2+y^2}}dx = \sqrt{x^2+y^2} + C(y)$$ $$\int3y\cos(3xy)dx = \sin(3xy) + C(y)$$ $$\int3x^2y^2dx = y^2x^3+C(y)$$
The final answer is:
$$u(x, y) = \sqrt{x^2+y^2} + \sin(3xy) + y^2x^3+C(y)$$
Is this correct?
Integration is a linear operation, so you can distribute it among addition/subtraction in general. That's perfectly legitimate.
This looks correct to me. Be sure to note that $C(y)$ may have a constant term, but I would even argue that the notation is sufficiently general as to make that clear. Nice work. If I were to nitpick, the symbol $C$ is so canonically associated with constants of integration that you might want to use a different symbol, but it is fine the way it is. And from the point view of the integrals, it is constant in $x$ anyway.