  Dirac's ket-bra formalism is the "language" of quantum mechanics and quantum
field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we
have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra
projectors. In this work by alternately using the technique of integration
within normal, antinormal, and Weyl ordering of operators we not only derive
some new operator ordering identities, but also deduce some useful integration
formulas regarding to Laguerre and Hermite polynomials. This opens a new route
of deriving mathematical integration formulas by virtue of the quantum
mechanical operator ordering technique.
