I am trying to verify the following formula involving Bessel functions of the first kind and am having no luck. The formula is
$ \int{\omega} J_n(\rho \omega)\mathrm d\omega = \frac {1} {\rho} \left\{ -\omega J_{n-1} (\rho \omega) + n \int{J_{n-1}(\rho \omega)\mathrm d\omega } \right\} $
I apologize if this is painfully obvious with integration by parts but I couldn't see it. Moreover, I get the impression from this other post about a nearly identical integral that the above may not be right.
Any help is greatly appreciated. Also, if there is a simpler way to express/solve this integral, I would also be very grateful for that.