New integration limits are calculated as:
\begin{equation} t=t_{0}\Rightarrow w=\frac{G}{\sqrt{t_{0}}},\ \ t=t_{f}\Rightarrow w=\frac{G}{\sqrt{t_{f}}}\nonumber \\ \end{equation}
The integral can now be integrated w.r.t \(w\), rather than w.r.t \(t\):