Bessel in AP?

Calculus Level 5

If

\[\large \int x(2J_{2}(x)-4J_{4}(x)+6J_{6}(x)-\ldots) \ dx=\frac{x^{A}}{B}J_{k}(x)+c\]

where \(A,B,k\) are integers and \(c\) is a constant of integration, find \(A^{B^{k}}\).


\(J_{n}\) are the Bessel functions of the first kind.

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