The aim of this paper is to generalize inequality f ( x ) - 1 b - a ∫ a b f ( s 1 ) d s 1 - f ( b ) - f ( a ) b - a x - a + b 2 - f ′ ( b ) - f ′ ( a ) 2 ( b - a ) · x 2 - ( a + b ) x + a 2 + b 2 + 4 ab 6 ⩽ ∥ f ‴ ∥ ∞ · ( b - a ) 3 6 I x - a b - a obtained in [A. Aglić Aljinović, M. Matić, J. Pečarić, Improvements of some Ostrowski type inequalities, J. Comput. Anal. Appl., in press], and therefore obtain a generalization and improvement of inequality f ( x ) - 1 b - a ∫ a b f ( s 1 ) d s 1 - f ( b ) - f ( a ) b - a x - a + b 2 - f ′ ( b ) - f ′ ( a ) 2 ( b - a ) · x 2 - ( a + b ) x + a 2 + b 2 + 4 ab 6 ⩽ ∥ f ‴ ∥ ∞ · A ( x ) ( b - a ) 3 obtained in [G.A. Anastassiou, Univariate Ostrowski inequalities, Revisited, Monatsh. Math. 135 (2002) 175–189]. To do this, first we derive general Euler–Ostrowski formulae which generalize extended Euler formulae, obtained in [Lj. Dedić, M. Matić, J. Pečarić, On generalizations of Ostrowski inequality via some Euler-type identities, Math. Inequal. Appl. 3(3) (2000) 337–353]. The main novelty is that a remainder is expressed in terms of B n ∗ ( x - mt ) which enables us to obtain a vide variety of quadrature formulae such as trapezoid, midpoint, bitrapezoid, twopoint formulae and their multipoint generalizations.