I know this is a very simple question but why is this wrong?$$\int(xdy+ydx)=\int xdy+\int ydx=x\int dy+y\int dx=2xy$$
I saw a similar question on Stack Exchange, but it was too complicated for me to understand. I am in 11th Grade and I have just done basic differentiation and integration for physics. Any help would be appreciated!
Since $x,\,y$ are in general not independent, you can't treat $x$ as a constant as in $\int xdy=x\int dy$. Your original problem would make this clearer if you wrote $x(y)dy+y(x)dx$. In fact,$$\int(xdy+ydx)=\int d(xy)=xy+C.$$