Asymptotically optimal definite quadrature formulae of 4th order

We construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants. Besides the asymptotical optimality, an advantage of our quadrature formulae is the explicit form of their weights and nodes. For the remainders of our quadrature formulae monotonicity properties are established when the integrand has a non-negative or non-positive fourth derivative, and a-posteriori error estimates are proven.