understanding basic Laplace transformation

Question

I am trying to understand Laplace transformations. Could someone tell me from where the fraction (1/-s), in red on the first line is originated?

https://i.stack.imgur.com/Xksa3.jpg

Answer 1

$\color{red}{\left(\dfrac{1}{-s}\right)}$ comes because you do $dt\color{red}{(-s)}$

You change one $1$ for and $s$ accidentally

edit other way: since $\left(\dfrac{e^{-st}}{-s}\right)'=e^{-st}$ then $\mathcal{L\{1\}}=\displaystyle\lim_{b\to\infty} \left( \dfrac{e^{-sb}}{-s}-\dfrac{e^{-s.0}}{-s} \right)$

Answer 2

the neutral factor $\frac{-s}{-s}$ has simply be introduced, then one part kept in the integral and the other out...