Peter: better results if you use Log and Sinh instead of log and sinh.
With these changes Mathematica says they don't converge except
the second, which is pi,
and the last which is looks like this depending on -1<Re(k)<0 or not..
If[Inequality[-1, Less, Re[k], Less, 0],
3^k*Pi*Csc[k*Pi]*(Pi*Cot[k*Pi] - Log[3]),
Integrate[(x^k*Log[x])/(3 + x), {x, 0, Infinity},
Assumptions -> Re[k] <= -1 || Re[k] >= 0]]
RJF
Peter Gustafson wrote:
>My knowledge of Mathematica syntax isn't great, but I think this is
>correct.
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