To solve integrals in the absolute nodal coordinate method and address the difficulty in applying it to an arbitrary-section beam, this paper focuses on two methods involving single integrals:the invariant matrix method and the Gerstmayr method, with cross-section characteristics by applying the interpolation of a discrete function. Such single integrals demonstrate that the nodal coordinate method can be applied to an arbitrary-section beam. The Euler-Bernoulli beam used in engineering structures is characterised by a symmetrical cross-section, small section size, zero odd integrals and negligible high-order even integrals, which simplifies the single integrals of the two methods. Finally, the Gaussian integration is adopted to improve the solving efficiency of elastic force and force Jacobian.