Gaussian integral via $F(t)^2+G(t)$ constant

For \(t\ge0\) let \[ F(t)=\int_0^t e^{-x^2}\,dx \] and \[ G(t)=\int_0^1\frac{e^{-t^2(1+x^2)}}{1+x^2}\,dx. \] Show that \(F(t)^2+G(t)\) is constant and deduce the value of \(F(\infty)\).