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

euphoricrhino
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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!
 
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