I have a problem: I'm working on a big-ish proof and it all relies on whether I can express the fraction of integrals as one integral. Any tips on how I can even attempt to find $\mathcal{G}(\rho,\pi)?$
$$\frac{\int_{x\in A}\frac{2\rho(x)h(x)}{\pi(x)}\,dx}{\sqrt{\int_{x\in A}\frac{\rho^2(x)}{\pi(x)}\,dx}}=\int_{x\in A}\mathcal{G}(\rho,\pi)h(x)\,dx.$$
Some background info: $\rho$ and $\pi$ are probability densities, h is an arbitrary functional (coming from taking directional derivative, along arbitrary direction) and I need $\mathcal{G}$ to solve for $\rho$ in
$0=L(\theta)+\dfrac{k}{2}\,\mathcal{G}(\rho,\pi)-\bar{\lambda} $
It would be enough if I can find $\mathcal{G}$ for this specific problem, but more formally I'm looking for a solution to $$\frac{\int\frac{df}{d\rho}\,dx}{\sqrt{\int f(\rho)\,dx}}=\int \mathcal{G}\,dx.$$
Also, worst case if closed form solution does not exists, it would be enough to find a way to compute the numerical value of $\rho(\theta_i)$, if I was given $\theta_i.$