In this paper, an alternative and integral method is proposed in order to build a formal slender–body theory valid up to high orders with respect to the slenderness ratio $\epsilon$ and for a non–lifting body which is not necessarily of circular cross–section. The method consists of asymptotically expanding and inverting the Fredholm integral equation of the second kind bearing on the unknown source density, which may be spread on the boundary of the body. For such a treatment, the concept of integration in the finite–part sense of Hadamard is powerful. The source density is then given up to order o(ϵ3) and the pressure coefficient is provided on the body up to order O(ϵ4 log ϵ). Throughout this paper, special attention is also paid to the main considered case of the axially symmetric slender body.