Laplace transform $ \mathscr{L} \left( \int_0^t \frac{e^{-\tau}-1}{\tau}d\tau \right) $

Question

Can anyone give me a hint to solve

$$ \mathscr{L} \left( \int_0^t \frac{e^{-\tau}-1}{\tau}d\tau \right) $$

Answer 1

First try finding the laplace transform of this function:

\begin{equation*} \frac{e^{-t} -1}{t} \end{equation*}

Using the rule:

\begin{equation*} L\left(\frac{f( t)}{t}\right) =\int ^{\infty }_{s} g( s) \end{equation*} where

\begin{equation*} L( f( t)) =g( s) \end{equation*} then try using the rule

\begin{equation*} L\left(\int ^{t}_{0} f( t) dt\right) =\frac{g( s)}{s} \end{equation*} I thin this should lead you where you want.