Is it possible to eliminate $g(x)$ in the formula $\dfrac{\int g(x)a(x)}{\int g(x)b(x)}$ I am hoping someone can show me how to do the following:
I have two integrals containing a function $g(x)$ that I would like
to eliminate.
The formula looks like this.
$\frac{\int g(x)a(x)}{\int g(x)b(x)}$ 
Is there any way to eliminate the $g(x)$ terms?
Thanks
Paul

Here's a simple counter example:
$$\begin{align}
g(x) &= x^k\\
a(x) &= x^m\\
b(x) &= x^n
\end{align}$$
Now
$$\frac{\int g(x)a(x)\,\mathrm dx}{\int g(x)b(x)\,\mathrm dx} = \frac{(k+m+1)^{-1}x^{k+m+1}+C_1}{(k+n+1)^{-1}x^{k+n+1}+C_2}$$
but
$$\frac{\int a(x)\,\mathrm dx}{\int b(x)\,\mathrm dx} = \frac{(m+1)^{-1}x^{m+1}+C_1}{(n+1)^{-1}x^{n+1}+C_2}$$
As you can see, not only does the presence of $g(x)$ make a difference, but the result also depends on the specifics of $g(x)$, like the value of $k$.
