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I am a geologist working on a fluid mechanics problem. Solving the PDE for my problem, this Bessel integral arises:

\int_{0}^{R} x^3 J_0 (ax) dx

where J_0 is the Bessel function of first kind, and a is a constant.

I haven't found the solution in any table or book, and due to my limited background in applied mathematics I don't know how to integrate it by myself.

Does anybody know the solution?

Thanks a lot in advance