How do you integrate a Bessel function? I don't want to memorize answers or use a computer, is this possible? 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.

I would do it numerically. Also for x not too large, you could use the power series expansion, but this will run into convergence issues before x gets too high. Numerical integration can be done to high accuracy without too much computation using Gaussian quadrature. Good luck
looking for an analytic solution (although maybe one exists?). Possibly you can use Bessel's equation, and by substituting your integral you can derive a new differential equation, but I think you'd be very lucky if this allowed an analytic solution. But, a numerical solution should be straightforward (at least for fixed N).
