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I am reading the Polymer Scattering wikipedia page (https://en.wikipedia.org/wiki/Polymer_scattering). I don't understand how using an isotropic assumption they have gone from the first equation to the last equation. I thought it might be a taylor series followed by some approximation but I have not been able to work it out.

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If the Hamiltonian does not depend on the angles (per def of isotropy), then one can simply integrate out the angular DOF. Try integrating over $d\, \cos \theta$ and then you get the result. Where $\theta$ is the angle between $k$ and $R$

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