I am looking for the solution to the integral:
$$\int_{a}^{\infty} x J_n(\alpha x)\;dx$$
where $a<< \alpha$ and $n$. I get something out of Mathematica for $a=1,2,0.1...$ in terms of the Hypergeometric function. It makes me think that some conditional expression for the solution exists, I just fail to find it.
The original integral of interest is actually this one
$$\int_{0}^{\infty} x J_n(\alpha x)\;dx$$
However, I know, this one does not converge. Maybe someone knows how to estimate it at least in certain limits (n should be any)... I need to extract something sensible, not just "you failed, go on with your life" kind of a situation...
I have tried to truncate it on top and take a limit to infinity, but that also doesn't work.
Your original integral is the Hankel Transform of order n of the function $f(x) = 1$.
The Hankel Transform of order n, and its inverse, can be defined as
$$\mathscr{H}_n\left\{f(r)\right\} = F(q) = \int_0^\infty f(r)J_n(qr)r\space dr$$ $$\mathscr{H}^{-1}_n\left\{F(q)\right\} = f(r) = \int_0^\infty F(q)J_n(qr)q\space dq$$
Note that the Hankel Transform is its own inverse.
For $n = 0$, and for an appropriate definition of the Dirac Delta distribution, it is easy to show the answer to your integral by using the inverse transform:
$$\mathscr{H}^{-1}_0\left\{\dfrac{\delta(q)}{q}\right\} = \int_0^\infty \dfrac{\delta(q)}{q}J_0(qr)q\space dq = 1$$ $$\mathscr{H}_0\left\{1\right\} = \int_0^\infty J_0(qr)r\space dr = \dfrac{\delta(q)}{q}$$
For higher orders of $n$, try to find a table of Hankel Transforms from a reliable source.