Hy karatekid.

Consider f(x) = sin(x) and g(x) = x from 2*pi to 4*pi. The Integral of f(x) = 0 over that region but now the integral of f(x)/g^2(x) is not zero:

http://www.wolframalpha.com/input/?i=int+sinx%2Fx^2+dx+from+2*pi+to+4*pi

Basically you can choose any function that scales the origin in such a way the symmetry of the parts that were balanced (i.e. add to zero) have become un-balanced with the right kind of scaling.