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I'm considering an expression of the form $$\int_{-\infty}^\infty dx G(x) \delta(x^2-f(x)^2) $$ where $G$ and $f$ are two unrelated smooth functions of $x$. Now I know that when $f$ is a positive constant, I can write this as $$ \int_{-\infty}^\infty dx \frac{G(x)}{2f} (\delta(x-f)+\delta(x+f)). $$

Now, f is a function of $x$ and I do not have an explicit form of the roots of $x^2-f(x)^2$, but I do have the implicit equations $x=f(x)$ and $x=-f(x)$ for the roots. Can I write my first expression in a form similar to the second equation? That is, I would like to split the Dirac delta in the first expression and end up with something like

$$ \int_{-\infty}^\infty dx H(x) \left(\delta(x-f(x))+\delta(x+f(x))\right). $$ where $H$ is some known function of $x$ (and $G(x)$ and $f(x)$).

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