I have an improper integral whose integrand consists of a Cosine term, exponential term and zero-order Bessel functions. I have tried using "Integrate" and "NIntegrate" but I get an un-converged output. I have also tried to convert the integral into a double integral using the approach mentioned in Chapter 9 of the book "Transforms and Applications Handbook",Poularikas but I still get an un-converged solution.
What would be the right approach to go about this integral ?
Numerically:
f[\[Alpha]_, t_, a_, b_] := NIntegrate[1/(2 \[Alpha])* Exp[-x^2/(4 \[Alpha])] Cos[x t]*(BesselJ[0, x*a] + BesselJ[0, x*b])*x, {x, 0, Infinity}] f[1, 2, 3, 4] (*0.115894*)
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