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I've been trying for a while now to find a method to compute this integral, $$\int_{\mathbb{R}^3}\frac{d^3\underline{k}}{(2\pi)^3}\frac{32\pi^2}{\kappa^4}|\underline{k}|\left(\kappa-\frac{|\underline{k}|}{3}\right)^2e^{-2\frac{|\underline{k}|}{\kappa}},$$ where $\kappa\in\mathbb{N}$ is a constant.

I've been thinking so far along the lines of integration by parts and using the fact the exponent will vanish at the boundary, but so far i've had no luck. Is this integral in a common form (or close to one)?

Thanks for the help.

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    $\begingroup$ Integrating over what? All of $\mathbb R^3$? $\endgroup$ Feb 18, 2015 at 21:44
  • $\begingroup$ What is $|\underline \kappa |$? Do you mean $\lfloor \kappa \rfloor$, this wouldn't make too much sense... $\endgroup$
    – AlexR
    Feb 18, 2015 at 21:47
  • $\begingroup$ Yes integration is over $\mathbb{R}^3$. $\kappa$ is a integer constant, $\underline{k}$ is a vector in $\mathbb{R}^3$ so $|\underline{k}|$ is the modulus of that vector. Sorry about the confusing notation of $k$ and $\kappa$. $\endgroup$
    – Rammus
    Feb 18, 2015 at 21:52

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Hint: Use spherical coordinates $d^3|\vec{k}| = |\vec{k}|^2d| \vec{k}|d \phi sin \theta d \theta$. The integral over the angles is trivial (results in $4 \pi$) and now you integrate over $|\vec{k}| \in [0, \infty] $. Another hint for the Integration over $|\vec{k}|$: Make the Substitution $|\vec{k}|=\frac{\kappa u}{2}$ and expand the binomial. Now You can use the Definition of the Gamma function

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