At the very least, ∫uJ2n(u)du for integer n is expressible in terms of Bessel functions with some rational function factors.
To integrate uJ0(u) for instance, start with the Maclaurin series:
uJ0(u)=u∞∑k=0(−u2/4)k(k!)2
and integrate termwise
∫uJ0(u)du=∞∑k=01(k!)2∫u(−u2/4)kdu
to get
∫uJ0(u)du=u22∞∑k=0(−u2/4)kk!(k+1)!
thus resulting in the identity
∫uJ0(u)du=uJ1(u)
For ∫uJ2(u)du, we exploit the recurrence relation
uJ2(u)=2J1(u)−uJ0(u)
and
∫J1(u)du=−J0(u)
(which can be established through the series definition for Bessel functions) to obtain
∫uJ2(u)du=−uJ1(u)−2J0(u)
and in the general case of ∫uJ2n(u)du for integer n, repeated use of the recursion relation
Jn−1(u)+Jn+1(u)=2nuJn(u)
as well as the additional integral identity
∫J2n+1(u)du=−J0(u)−2n∑k=1J2k(u)
should give you expressions involving only Bessel functions.
On the other hand, ∫uJν(u)du for ν not an even integer cannot be entirely expressed in terms of Bessel functions; if ν is an odd integer, Struve functions are needed (∫J0(u)du cannot be expressed solely in terms of Bessel functions, and this is where the Struve functions come in); for ν half an odd integer, Fresnel integrals are needed, and for general ν, the hypergeometric function 1F2(bac∣u) is required.
This post imported from StackExchange Mathematics at 2014-06-12 20:40 (UCT), posted by SE-user J. M.