# Why NIntegrate fails to evaluate this integral?

Posted 2 months ago
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 Hi, I am trying to use NIntegrate to evaluate the following integral (known as one of the Jaeger integrals, which are certainly convergent): NIntegrate[F[x],{x,0,Infinity}] where F[x_]=Exp[-x]/x/(BesselJ[0,Sqrt[x]]^2+BesselY[0,Sqrt[x]]^2). Unfortunately, NIntegrate notoriously returns error messages suggesting the lack of convergence, either at x=0 or at x=infinity. I cannot figure out a satisfactory combination of integration options. Is there any way to overcome this problem? Lesław.
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Posted 2 months ago
 Leslaw: I think you mean "Infinity" not "infinity". Capitalization matters in the Wolfram Language. But, being symbolic means that you do not always get a helpful error message.Have a great and safe holiday.
 Overcome this situation is substitute x=1/t^2 then plot and integral is:  Plot[{(2 E^(-(1/t^2)))/( t (BesselJ[0, 1/t]^2 + BesselY[0, 1/t]^2))}, {t, 0, 20}, PlotRange -> All] NIntegrate[(2 E^(-(1/t^2)))/( t (BesselJ[0, 1/t]^2 + BesselY[0, 1/t]^2)), {t, 0, Infinity}, Method -> "GlobalAdaptive", WorkingPrecision -> 100] (*4.85471500790454846564370835036737574863704130040306375617123990141541\ 1155533681789950223687667641287*) NIntegrate[(2 E^(-(1/t^2)))/( t (BesselJ[0, 1/t]^2 + BesselY[0, 1/t]^2)), {t, 0, Infinity}, Method -> "GaussKronrodRule", MaxRecursion -> 40, WorkingPrecision -> 100] (*4.85471500790454846564370835036737574863704130040306375617123990141541\ 1155533681789950223687667641287*)