We can clean that up a bit if we make the assumption that t is real and greater than 0.
Assuming[t > 0, Integrate[BesselJ[d/2, x]/x, {x, t, \[Infinity]}]]
Integrate[BesselJ[d/2, x]/x, {x, t, \[Infinity]}]
returns
ConditionalExpression[
2/d - 2^(-1 - d/2) t^(d/2)
Gamma[d/
4] HypergeometricPFQRegularized[{d/4}, {1 + d/4, 1 + d/2}, -(t^2/
4)], Re[t] > 0 && Im[t] == 0]
Integrate[BesselJ[1, x]/x, {x, t, \[Infinity]}]
See you later, I need to go offline.
]]>ConditionalExpression[
1/2 (2 + BesselJ[1, t] (2 - \[Pi] t StruveH[0, t]) +
t BesselJ[0, t] (-2 + \[Pi] StruveH[1, t])),
Re[t] > 0 && Im[t] == 0]
Here t = |k| so both conditional expressions are automatically satisfied. Unfortunately I can't see a nice way of bounding those Struve functions without getting a power of |k| that is too large.
]]>Integrate[BesselJ[1, x]/x, {x, 0, \[Infinity]}]
The first integral seems to be close to 1 for |k| small and close to 0 for |k| large.
How did you determine that?
]]>