Thanks for your reply Villyer. I've just solved it, with some help from a friend.
Here's the solution.
For x \approx 0 , (\frac{1 - x}{x})^k = x^{-k} . Therefore, f(p) = \int_0^p x^{-k} dx = \frac{1}{1-k} p^{1 - k} , as required.
Cheers!
This problem arises in a paper on population genetics (Kimura 1962).
1. The problem statement
Let f(p) = \int_0^p ((1 - x)/x)^k dx.
For a small value of p, we have approximately
f(p) = (p ^ (1-k)) / (1-k)
How is this obtained?
2. My attempt at a solution
I tried to expand the f(p) around p =...
What Lagrangian would you set up?
We want to find the path that minimizes W = integral F dot dr. How do you solve for a path?
I apologize for the sloppy notation; this is my first post and I'm not familiar with tricks to get the integral and dot product signs to show up.