Why NIntegrate fails to evaluate this integral?

Questions
,
1

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.

2

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.

3

No, in my test code I used the infinity symbol. Lesław

4

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*)