I am wondering how to do the integral $$\int_{0}^{k} u J_n^2({u}) du, \ n \in \mathbb{Z}^+,$$ and express my answer in terms of other first kind Bessel functions. I have searched here for useful identities, but I can't seem to combine them in a way that helps.
Integration by parts hasn't proved to be useful either, since from what I've tried, I have to evaluate either $\int u J_{n}(u) du$ or $\int J_{n}(u)^2 du$, which both can't be expressed in terms of Bessel functions alone.
Does anyone have any hints? Thanks :-)
Hint:
Consider $$\dfrac{d}{dx}\left[x^2\left(J_{n}^2(x)-J_{n+1}(x)J_{n-1}(x)\right)\right]$$ and prove $$\int_{0}^{k} u J_n^2({u}) du=\dfrac12k^2\left(J_{n}^2(k)-J_{n+1}(k)J_{n-1}(k)\right)$$