i need to calculate this integral from plus to minus infinity

in this area

i need to find the points which turn to zero in the denominator

and non zero in the numerator.

i got two such points

by using this formula

the third point is z=0 but for it we have both numerator and denominator 0

i calculated the residium for it by formula

but then

my prof says some stuff that involves the area

he says that my points are 0 +1 -1 +2^(0.5) -2^(0.5) etc.. because the denominator goes to zero

for each point have a residiu and i need to sum the residiums inside.

but here the area is not defined

its not like (by radius 3)

i dont know what point are inside the area