I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various Schaum's) for this problem. I would like to do this by hand. Is it possible? I have had no luck with expanding out $J_n(x)$ and integrating term by term, as I cannot collect them into something nice at the end.
If possible and I just need to try harder (i.e. other methods or leaving it alone for a few days and coming back to it) that is useful information.
Thanks to anyone with a clue.