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Original Message (ID '165119') By Michael:
The problem might be a typo: Should be BesselJ instead of Besselj. By default, computation time is unconstrained. (You can constrain it, if you should ever want to, with TimeConstrained.) For your first integral I got In[2]:= Integrate[ k^2*Exp[I k h]*BesselJ[0, k g], {k, 0, \[Infinity]}] // Timing Out[2]= {49.125881, ConditionalExpression[( I Sqrt[1 - g^2/h^2] h (g^2 + 2 h^2))/(g^2 - h^2)^3, Abs[Im[g]] < Im[h]]} For the second, Mathematica did not come up with an answer, either with or without the assumptions I stuck in. In[8]:= Block[{$Assumptions = h \[Element] Reals && 0 < k1 < k2 && g \[Element] Reals}, Integrate[k^2*Exp[I k h]*BesselJ[0, k g], {k, k1, k2}] ] // Timing Out[8]= {15.322764, \!\( \*SubsuperscriptBox[\(\[Integral]\), \(k1\), \(k2\)]\(\( \*SuperscriptBox[\(E\), \(I\ h\ k\)]\ \*SuperscriptBox[\(k\), \(2\)]\ BesselJ[0, g\ k]\) \[DifferentialD]k\)\)} (When Mathematica returns the integral unevaluated, it means it could not find an answer in terms of standard functions.)