The nature of Boxer-Thaler and Madwed integrators is explored in this note. A consistent derivation of the Madwed integrator from the wellknown derivation of the Boxer-Thaler integrator is first proposed. A new general computerized algorithm is also proposed for the kth-order Boxer-Thaler and Madwed integrators. These two discrete integrators are used in this note to replace the Tustin integrator for digitizing a continuous-time system. A more systematic and precise formulation of the Q-matrix is presented for the s-domain to z-domain transformation via Boxer-Thaler and Madwed integrators. Due to the more accurate nature of these two discrete integrators, better results can be obtained. A set of MATLAB programs is written to implement the proposed algorithms in this note.