Trying to prove a property for the region of convergence of a Laplace transform

I've been studying Schaum's outlines for Signals and Systems, and came across this: And I understood the part until $$\int_{t_{1}}^{t_{2}} |x(t)|e^{-\sigma_{0}}e^{-(\sigma_{1}-\sigma_{0})t}\,dt$$

Where did the $$-\sigma_{0} -(\sigma_{1} - \sigma_{0})t$$ come from? Did the book add/divide the process? It didn't really explain what to do before that, so I might have overlooked something.

I got that the exponential with complex values is equal to one because of the absolute value, but after that, I get a bit lost by the addition of another exponential

• I'm not sure that's exactly your question, but they just wrote $\sigma_1 t =(\sigma_1 -\sigma_0 +\sigma_0) t$ and then used the properties of the exponential Nov 25 '21 at 10:04
• What's the notion or purpose of equating $\sigma_{1}t$ to $( \sigma_{1} - \sigma_{0} + \sigma_{0})t$ though? I don't really know how it links up to proving that the ROC of X(s) becomes the entire s-plane after that Nov 25 '21 at 11:38