Int(dxdy*xe^(-x-xy)) = Int(dx*xe^(-x) * Int(dy*e^(-xy)) )
where I have split the integral into components.
The second integral (the one you are interested in) can be simply done by assuming x is a constant (it doesn't vary as y varies anyway), so
Int(dy*e^(-xy)) = 1/(-x) * e^(-xy)
where I have neglected the integration constant.