\[\int_{0}^{x}J_{n}\left(t\right)J_{n+1}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}%
\left(1-{J_{0}}^{2}\left(x\right)\right)-\sum_{k=1}^{n}{J_{k}}^{2}\left(x%
\right)=\sum_{k=n+1}^{\infty}{J_{k}}^{2}\left(x\right),\]