hi i'm working through a past exam problem and i dont have any solutions i was wandering if some one could tell me if im doing this correctly

we have the real integral from negative infinity to possitive infinty of

1/(x^4 +1)

so to work this out i work in terms of z instead of x.

factorise z^4 +1 into its linear factors. which are:

e^(pi/4), -e^(pi/4), e^(-pi/4), -e^(-pi/4)

then looking at where the singularites occur we notice that two are in the upper half of the complex plane and two are in the lower half.

therefore the integrand = 2 the integral from 0 to positive infinity

we then find the residues of the two singularities and and then apply the residue theorem

so im pretty good up here, but then finding the residues i got a bit lost with all the exponentials and positives and negatives.

so if anyone could just help me out it should be all good.

thanks