Question

Playing around with integrals, I'd like to find out if this is right or not but not sure how to check it.

Answer 1

\[\int_{-\infty}^\infty \frac{dx}{(x^2+p^2)^{k+1}} = \frac{2 \pi}{(2p)^{2k+1}} \binom{2k}{k} \]

Answer 2

looks this can be turned into a beta integral

Answer 3

Oh that's a good idea, you're probably right about that.

Answer 4

letting \(x = p\tan\theta\) :

\[\int_{-\infty}^\infty \frac{dx}{(x^2+p^2)^{k+1}} =\dfrac{2}{p^{2k+1}} \int_0^{\pi/2} \cos^{2k}\theta\, d\theta\]

Answer 5

that integral is a beta function \(\dfrac{1}{2}B(k+\frac{1}{2},\frac{1}{2})\)

Answer 6

http://gamma-function.netne.net/img379.png