In this paper, we propose a new integral representation for reconstructing the surface density of matter in the flat discs of spiral galaxies. The surface density is expressed through the observed rotation velocity curves of visible matter in the discs of spiral galaxies. The new integral representation is not based on the quadrature of special functions. The solution that is found is used to process and analyse observational data from several spiral galaxies. The new integral representation can be used to more accurately estimate the amount of dark matter in spiral galaxies.