Integral 1/(x^k +1) from 0 to infinity

[Guest]

How do I integrate 1/(x^k + 1) from 1 to infinity where k is any real number such that k>1?

This the list of standard definite integrals on this site gives:

pi / (k * sin(pi / k))

However, k is labelled n, which suggests that the result may hold only for integer k. The graph of this function looks sensible for fractional k, but this function is horrible to integrate numerically, so I have not been able to check it accurately.

Any help will greatly appreciated!