Existing works on orthogonal moments are mainly focused on optimizing classical orthogonal Cartesian moments, such as Legendre moments, Gauss-Hermite moments, Gegenbauer moments, and Chebyshev moments. Research in this area generally includes accurate calculation, fast computation, robustness/invariance optimization, and the application of orthogonal moments. This paper presents the inclusion of the integration method proposed by Holoborodko to calculate the Legendre moments. The results obtained are compared with the traditional equation and the methods proposed by Hosny and Pawlak to approximate the integration computation.