It is well-known that the remaining term of any n-point interpolatory
quadrature rule such as Gauss-Legendre quadrature formula depends on at least
an n-order derivative of the integrand function, which is of no use if the
integrand is not smooth enough and requires a lot of differentiation for
large n. In this paper, by defining a specific linear kernel, we resolve this
problemand obtain new bounds for the error of Gauss-Legendre quadrature
rules. The advantage of the obtained bounds is that they do not depend on the
norms of the integrand function. Some illustrative examples are given in this
direction.