Dividing an integrand by a strictly positive function
I've got a really nasty integral that I want to show is zero. It's of the form (x,y are functions of time)
I'm thinking I need to use Green's Theorem and the algebra for this would be massively simplified if I could discard the denominator, which is positive everywhere. So my question is, is the following true? How would I show it, and why not if it isn't?
Re: Dividing an integrand by a strictly positive function
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:
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.