Integration of Bessel function products (J_1(x)^2/xdx)

[euphoricrhino]

Hello,
While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral
$\int_0^\infty J_1(x)^2\frac{dx}{x}=1/2$

I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations, series representation of J_n, etc) but have not succeeded.

Can anyone provide a sketch of the proof?
Thanks!

 

[Charles Link]

See https://www.physicsforums.com/threa...n-using-bessel-functions.916241/#post-5775639
It comes up in diffraction theory with the airy disc. I do not know how to compute it though.