For this identity of the delta function:
$$
\int\_{-\infty}^\infty f(x)\delta(g(x)) = \sum\_{n = 1}^N\int\_{-\infty}^\infty dx f(x)\frac{\delta(x-x\_n)}{|g'(x\_n)|}
$$

Where $x\_n$'s are zeros of $g(x)$. Does it still hold if the integrand goes from $0$ to $\infty$ instead?

Thanks!