The absorbed intensity of microwaves penetrating a quasi 1d spin chain compound, as measured e.g. in ESR experiments, is determined by the imaginary part of the dynamical susceptibility of the compound. The latter is hard to calculate over the full range of experimentally available values of temperature and magnetic field, even in for the integrable 1d Heisenberg chain. Theoretically more accessible quantities are the moments of the normalized intensity. As suggested by Maeda, Sakai and Oshikawa in 2005 the first two moments determine the resonance shift, measured in ESR experiments on systems that can be seen as small perturbations of the Heisenberg chain. Combining perturbation theory, high-temperature expansions and scaling arguments with exact results I shall discuss applicability and limitations of a general method of moments applied to close to isotropic systems. The resulting insight will then be used for a detailed analysis of ESR and thermodynamic data for the spin chain compound Cu(py)₂Br₂ (CPB), where (py) stands for the pyridine molecule NC₅H₅. The analysis allows us to fully and precisely determine the g-tensor and to identify directions and magnitude of the magnetic anisotropies of this compound..