The Question
The inverse Laplace Transform formula can be read quite intuitively: $$ x(t) = \frac{1}{2\pi j}\int^{\sigma + j\infty}_{\sigma - j\infty} X(s)e^{st} ds $$ Given a particular damping factor $\sigma$, the signal in time domain is a weighted sum of eigenfunction $X(s)e^{st}$, across all frequency. The $\frac{1}{2\pi j}$ is just an overall scaling factor.
Now, how can I go from the inverse Laplace Transform formula to the Laplace Transform formula? (for example, the Bilateral Laplace Transform formula?) $$ X(s) = \int^{\infty}_{-\infty}x(t)e^{-st}dt $$
Supplementary Information
(Thank you Matt L. for answering this part)
I have tried to plug the Bilateral Laplace Transform formula into the Inverse Laplace Transform to see if I can obtain some insight, but after writing the initial step down, I don't know how should I proceed. $$ x(t) = \frac{1}{2\pi j}\int^{\sigma + j\infty}_{\sigma - j\infty} \left( \int^{\infty}_{-\infty}x(t)e^{-st}dt \right) e^{st} ds $$