I am doing research in population genetics. Trying to understand one of the journal articles I am reading, I just don't get how the author moved from one equation to the other.

Here is what I have:

k = [1+u(1+s)n/(N-n)]/[(1+s)(1-u)+r]; is a function of n

In theory, the probability of the process that I am working on should be:

P = 1/[1+Sigma(i=1 to N-1) Product(n=1 to i) k]

ie, P = 1/[1+Sigma Product k]

So replacing the function k in the probability equation P, and assuming that u, s and r are <<1, the author obtains:

value of P is approximate to 1/[N* Integral (between 0 and (1-1/N) of [(e^(-xN(s+r))*((1-x)^(-uN))*dx].

I just don't get where the author gets the x variable from. Did he do some change of variable. If so, what is the function x and how did the author move between the theoretical value of the probability and the calculated approximation?

Been cracking my head all day over this and I still can't figure it out. I tried using Mathematica but plugging k into the P equation gives 10 lines of text, with Gamma functions and stuff like that. I'm totally lost now and need help to move on with my research.

Thank you in advance