Dividing an integrand by a strictly positive function

Oct 2010
Hi all,

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)

\(\displaystyle \oint_C{\frac{f(x,y) dx + g(x,y) dy}{\dot{x}^{2} + \dot{y}^{2}}\)

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?

\(\displaystyle \int{f dx} = 0 \Rightarrow \int{\frac{f}{g^2} dx} = 0\)


MHF Helper
Sep 2012
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:


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.