The article deals with quasi-interpolants of a function \(f\), \[ Qf = \sum_{i \in \mathbb Z} \lambda f (\cdot + i) \phi(\cdot - i), \] where \(\phi\) is a certain piecewise polynomial partition of unity with a compact support, and where \(\lambda\) is a linear functional of the form \[ \lambda f=\sum_{j \in J} \gamma_j \langle f , \psi(\cdot + j) \rangle. \] Here \(J\) is a finite subset of \(\mathbb Z\), \(\gamma_j\) are the weights, \(\psi\) is another piecewise polynomial partition of unity with a compact support, and \(\langle \cdot, \cdot \rangle\) is the usual inner product. These are the integral quasi-interpolants (iQIs). The authors characterize the exactness of an iQI in a subspace of polynomials (reproducing property) and derive an expression for the error. The optimal weights \(\gamma_j\) are chosen to minimize a relevant term in the right hand side of the upper bound of this error under assumption of the reproducing property. The existence of the solution of this problem is proven. Additionally, explicit solutions for the case when \(\phi\) is given by B-splines are presented. The conclusions are illustrated by numerical experiments.