  We present the results of searches for high-energy muon neutrinos from 41
gamma-ray bursts (GRBs) in the northern sky with the IceCube detector in its
22-string configuration active in 2007/2008. The searches cover both the prompt
and a possible precursor emission as well as a model-independent, wide time
window of -1 h to +3 h around each GRB. In contrast to previous searches with a
large GRB population, we do not utilize a standard Waxman-Bahcall GRB flux for
the prompt emission but calculate individual neutrino spectra for all 41 GRBs
from the burst parameters measured by satellites. In none of the three time
windows do we find a deviation from the background-only hypothesis. Therefore,
we place 90% CL upper limits on the fluence from the prompt phase of 3.7x10^-3
erg cm^-2 (72 TeV - 6.5 PeV) and on the fluence from the precursor phase of
2.3x10^-3 erg cm^-2 (2.2 TeV - 55 TeV), where the quoted energy ranges contain
90% of the expected signal events in the detector. The 90% CL upper limit for
the wide time window is 2.7x10^-3 erg cm^-2 (3 TeV - 2.8 PeV) assuming an E^-2
flux.
