Confusion in proof of Inverse Laplace Transform
To summarize the discussion:


The usual substitution is $s = \sigma + j \omega$ where $\sigma$ is the real part of the $s$ variable and $\omega$ is the imaginary part.
The equation in the image is for the Fourier transform, not the Laplace transform. The Fourier transform can be thought of as the Laplace transform evaluated on the imaginary axis ($\sigma = 0$).
The differential $ds$, when looking at real and imaginary parts distinctly, becomes $d\sigma + j d\omega$.
Any differential is an infinitesimal (very small) change in that variable. $dx$ is a small change in $x$.