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.
One can set $\rho=1$ without loss of generality. According to this page (see the paragraph $p+1$ dependency), $$ \omega J_{n}(\omega)=(n-1)J_{n-1}(\omega)-\omega (J_{n-1})'(\omega)=-(\omega J_{n-1}(\omega))'+nJ_{n-1}(\omega). $$ Hence a primitive of $\omega J_{n}(\omega)$ is $-\omega J_{n-1}(\omega)$ plus $n$ times a primitive of $J_{n-1}(\omega)$. This is your formula.