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Can someone explain to me how Int[0,∞](t^{n+1} x^n e^{-tx} dx) = Int[0,∞](x^n e^{-x} dx)? That is, the last two equations of section 1.

Let t -> 1 which you can do because this entire trick is about interchanging limits (since both an integral and a derivative are limits)

Ah, thanks! So it's just "has to hold for all values of t > 0, including for t = 1"