Hi,
I would tend to think that too.
Contour integration would probably mean considering f to be a complex variable with real and imaginary parts:
f=a+b*i
You'd have to look into how to do that for this problem.
If you try different values for T in the first problem you had, you quickly see that 1/abs(T) is probably right and of course for values that are only positive 1/T is probably right. So if you do the same for your new problem you should see the same. This works on a try by try basis because it's easier to integrate when T or Fc is a constant numerical value like 1,2,3,... etc. So it should work for every positive value tried for T or Fc after you replace it and then do the integration by your usual method. Granted this isnt a definite proof, but may still be useful for a given range of the variable.