[VA] Short & Sweet Math Challenge #25 "San Valentin's Special: Weird Math"

[Werner]

The key to the value of the integral in #3 is to realize that the integral is symmetric in the integral boundaries. Basically it is

integral(a,b,f(a+b-x)/(f(x) + f(a+b-x)),dx)

if you substitute y = a+b-x, dy=-dx, you get

integral(b,a,-f(y)/((y) + f(a+b-y)),dy) = integral(a,b,f(y)/((y) + f(a+b-y)),dy)

Meaning the value of the integral is the sum of both divided by two, or (b-a)/2, whatever f(x) is. So here it is (φ-1)/2.

Cheers, Werner