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You don't even need to numerically integrate. Each of your intended integrals is simply: $$\int_0^\infty e^{-k x^2}dx={1\over 2}\sqrt{\pi\over k}$$ Also you don't need to evaluate a bunch of Bessel functions, since BesselJZero[1/2,n] is $n\pi$. As noted by @belisarius, your first term would diverge if you integrate to $\infty$, since the integrand is 1. ...