When you have to integrate a function that requires substitution and you integrate it again, why is it wrong to keep the initial substitution?
e.g. $y''=\frac{2x}{(1+x^2)^2}$
If you let $u=1+x^2$ then $y'=-(1/u)+C$. Why is it wrong to integrate that again with respect to $u$ and then change back to $x$ at the end? I know it's not right but I can't see why