Note on certain integrals involving Bessel functions

The subject of this paper is to give some infinite integrals of the type
$$ \int\limits_0^\infty J_\nu(t)\{e^{\frac{\nu\pi i}2}K_{2\nu}(2\varepsilon\sqrt{xt})+e^{-\frac{\nu\pi i}2}K_{2\nu}(2\varepsilon\sqrt{xt})\}\,dt=K_\nu(t), $$
where $J_\nu(t)$, $K_\nu(t)$ denote Bessel functions.