\begin{equation}
\mathscr{L}[f(t)] =F(s) = \int\limits_{0}^{\infty}e^{-st}f(t)\,dt\tag{6.3.3}
\end{equation}
Integrating equation (6.3.3) w.r.t. ’s’, we get -
\begin{equation*}
\int\limits_{s}^{\infty}F(s)\,ds = \int\limits_{s}^{\infty}\left[\int\limits_{0}^{\infty} e^{-st}f(t)\,dt\right]\,ds
\end{equation*}
\begin{equation*}
= \int\limits_{0}^{\infty} \left[\int\limits_{s}^{\infty} e^{-st}f(t)\,dt\right]\,ds = \int\limits_{0}^{\infty}\left[\frac{ e^{-st}f(t)}{-t}\right]_{s}^{\infty} \,dt
\end{equation*}
\begin{equation*}
=\int\limits_{0}^{\infty}\frac{f(t)}{-t}\left[e^{-st}\right]_{s}^{\infty}\,dt = \int\limits_{0}^{\infty}\frac{f(t)}{-t}\left[0-e^{-st}\right]\,dt
\end{equation*}
\begin{equation*}
= \int\limits_{0}^{\infty}e^{-st}\{\frac{1}{t}f(t)\}\,dt = L\left[\frac{1}{t}f(t)\right]
\end{equation*}
i.e.,
\begin{equation*}
\mathscr{L}\left[\frac{1}{t}f(t)\right] = \int\limits_{s}^{\infty}F(s)\,ds
\end{equation*}
