While working on a flow problem, i came across non-trivial dual integral equations. For their solution, ome must deal with the following infinite integral that involves a combination of Bessel and trigonometric functions: $$ \int_0^\infty \lambda \left( I_0(\lambda) K_0(\lambda) - I_1(\lambda) K_1(\lambda) \right) \sin (\lambda z) J_0(\lambda t) \, \mathrm{d}\lambda \, , $$ wherein $J_0$ denotes the zeroth-order Bessel functions of the first kind, and $I_n$ and $K_n$ are, respectively, the $n$th order modified Bessel function of the first kind and the $n$th order modified Bessel function of the second kind.
In addition, i must evaluate the closely related integral $$ \int_0^\infty \lambda \left( I_0(\lambda) K_0(\lambda) - I_1(\lambda) K_1(\lambda) \right) \cos (\lambda z) J_1(\lambda t) \, \mathrm{d}\lambda \, , $$ with $J_1$ denoting the first-order Bessel function.
i was wondering whether a closed analytical expression for the above integrals is possible. This is important to work on the next step of the calculations.
Any help or hints are highly appreciated and desirable!
Thanks and bests, A