Can I say $ iint f (x, y) dx dy = int f (x) dx cdot int f (y) dy $ usually? ….

I can say that $$ iint f (x, y) dx dy = int f (x) dx cdot int f (y) dy $$ usually ? Is it permissible to divide the integral in that way? And we know that integtal is linear. $$ int f (x) + g (x) dx = int f (x) dx + int g (x) dx $$ I think this is not possible for $$ iint f (x) + g (x) dx = int f (x) dx + int g (x) dx $$ straight ? On top of that, why is it $$ iint f (x) g (y) dx dy = int f (x) dx int g (y) dy $$