numerical integration - highly oscillatory integral

Hi am trying to compute the following integral in Mathematica:

    NIntegrate[(Sign[Cos[q]] Sqrt[Abs[Cos[q]]])/(q+100),{q,0,Infinity}]

But when I run that command I get the following error:

    NIntegrate::ncvb: NIntegrate failed to converge to prescribed 
    accuracy after 9 recursive bisections in q near {q} = {6.3887*10^56}.
    NIntegrate obtained -317.031 and 171.60663912222586` for the integral
    and error estimates.

Is there a way to solve this issue? Thanks

Answer

Here's a manual implementation of the extrapolating oscillatory strategy. NIntegrate balks at trying it because the cosine is buried inside other functions.

ClearAll[psum];
psum[i_?NumberQ, opts___] := 
 NIntegrate[(Sign[Cos[x]] Sqrt[Abs[Cos[x]]])/(x + 100),
  {x, (i + 1/2) π, (i + 3/2) π}, opts];

res = NSum[psum[i], {i, 0, ∞},
    Method -> {"AlternatingSigns", Method -> "WynnEpsilon"},
    "VerifyConvergence" -> False] +

  NIntegrate[(Sign[Cos[x]] Sqrt[Abs[Cos[x]]])/(x + 100), {x, 0, (1/2) π}]
(*
  0.00010967(-)
*)

This agrees with Chip Hurst's result to almost 5 digits. But the method is reasonably quick, so we can examine the convergence as the WorkingPrecision (and therefore the PrecisionGoal) is raised:

ClearAll[res];
mem : res[wp_: MachinePrecision] := mem =
  NSum[psum[i, WorkingPrecision -> wp], {i, 0, ∞}, 
     Method -> {"AlternatingSigns", Method -> "WynnEpsilon"}, 

     "VerifyConvergence" -> False, WorkingPrecision -> wp] + 
   NIntegrate[(Sign[Cos[x]] Sqrt[Abs[Cos[x]]])/(x + 100),
     {x, 0, (1/2) π}, WorkingPrecision -> wp];

Table[res[wp], {wp, 16, 40, 8}]
(*
  {0.0001096696058468,
   0.000109676059727856341260, 
   0.00010967605972782927874728238851, 
   0.0001096760597278292787470847687426733221}

*)

% // Differences
(*
  {6.4538811*10^-9, -2.7062513*10^-17, -1.9761977*10^-25}
*)

At each step we extend the number of digits that agree, and we can see that the result seems reliable.

---

Alternatively, one can multiply and divide by cosine, which enables NIntegrate to use the "ExtrapolatingOscillatory" strategy. It's a bit slower because it introduces (integrable) singularities in the non-oscillatory part of the integrand, but again we get similar results.

ClearAll[res2];
mem : res2[wp_: MachinePrecision] := mem =
  NIntegrate[Cos[q] (1/Sqrt[Abs[Cos[q]]])/(q + 100), {q, 0, Infinity},
    Method -> "ExtrapolatingOscillatory", 
    MaxRecursion -> Ceiling[10 + 1.5 wp],  (* increase in MaxRecursion for singularities *)
    WorkingPrecision -> wp]

res2[]
(*
  0.000109673

*)

Table[res2[wp], {wp, 16, 40, 8}]
(*
  {0.0001096731867831,
   0.000109676059727866251185, 
   0.00010967605972782927874731895632, 
   0.0001096760597278292787470847687424806474}
*)