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The problem asks for a proof of the following equation, and I have no idea on how to approach this:

$$\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E) ,$$ for an arbitrary function $f(E)$, where $x$ and $p$ are the position and momentum, and $\delta$ is a Dirac delta function.

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Use:

$\delta(f(x))=\sum_{i} \frac{1}{\mid\frac{df}{dx}(x_i)\mid} \delta(x-x_i)$

where $x_i$ is the i_th zero of f(x)

i think you will find the result after doing this

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