I found the following link at math.com

http://www.math.com/tables/expansion/log.htm
I derived the first expression in the link and the one you gave the following way:

1/1-x = 1 + x + x^2 + x^3 + ...

let x = -x and integrate

1/1+x = 1 - x + x^2 - x^3

int {1/1+x dx} = x - x^2/2 + x^3/3 - x^4/4 ... = ln(1+x)

let x = x-1

ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 ... 0 < |x| < 1

The above expression is invalid at x=0.

How are the three expressions following the first in the link derived?