Integrate a logarithmic derivative directly

$\int(dy/dx)\,dx=\int2x/(x^2+1)\,dx$, hence $\int1\,dy=\int2x/(x^2+1)\,dx$. Therefore $y=\ln(x^2+1)+C$, where C is an arbitrary real constant. Since $x^2+1>0$, this solution is defined for every real x.