Digital integer and fractional order integrators and differentiators are very important blocks of digital signal processing. In many situations, integer order integrators and differentiators are not sufficient to model all kind of dynamics. For such systems, fractional order operators give better solution. This paper is based on design of a new family of fractional order integrators and differentiators using various approximation techniques. Here, digital fractional order integrators are designed by direct discretization method using different techniques like continued fraction expansion, Taylor series expansion, and rational Chebyshev approximation on the transfer function of Jain-Gupta-Jain second order integrator. Their response in frequency domain is compared. The frequency response of the proposed integrators with highest efficiency is also compared with the existing ones. It is proved that rational Chebyshev approximation based integrators have highest efficiency among them. The fractional order differentiators are also designed using proposed integrators. It is concluded that proposed family of fractional order operators show remarkable improvement in frequency response compared to all the existing ones over the entire Nyquist frequency range.