In this paper, new designs of infinite impulse response digital integrators are presented. Transfer functions of the digital integrators are derived after utilising the concept of multirate technique in the fractional interpolation of the rectangular and bilinear transform. Thereafter, the unknown variables of the obtained generalised transfer functions are optimised by using the optimisation algorithm. This yields the mean relative magnitude error, 63:439 dB and 78:771 dB for the first- and second-order, respectively. Furthermore, new designs of the first- and second-order digital differentiators are obtained by inverting the generalised transfer functions of the proposed integrator designs followed by optimisation of the unknown variables. The mean relative magnitude errors for first- and second-order differentiators are obtained as 56:478 dB and 75:095 dB, respectively. The proposed designs of integrators and differentiators exhibit the precise approximation of the ideal integrator and differentiator over the full Nyquist interval.