ABSTRACT
This chapter discusses in detail a variety of integrals based on the first and second order symmetric derivatives and on the approximate symmetric derivative. To some degree this parallels the story of integrals based on the ordinary derivative, but with interesting differences. It presents an account of a variety of symmetric integrals arising directly from these symmetric derivatives. The best known to date has been the James integral based on the second symmetric derivative and defined by a Perron method. There have been definitions of symmetric integrals as limits of Riemann sums. Preiss and Thomson produce an integral based on the approximate symmetric derivative in this way that integrates all everywhere convergent trigonometric series.
