This paper is concerned with the approximation, for a given bandlimit \(c>0\) and accuracy \(\varepsilon >0\), of integrals of the form \[ u(x)=\int _{-1}^1 e^{ictx} d\mu (t), \] where \(d\mu (t)=w(t) dt\) is a measure, with the sum \[ \overline u(x)=\sum_{k=1}^{M(c,\varepsilon)}w_ke^{ic\theta _kx}. \] The authors develop a method for constructing optimal nodes \(\theta _k\) and weights \(w_k>0\) such that \(|u(x)-\overline u(x) |\leq \varepsilon \), for \(x\in [-1,1]\). For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. The new fast algorithm presented can be extended to the construction of optimal nodes and weights for integrals involving prolate spheroidal wave functions, Bessel functions and other bandlimited functions. Various examples are presented an discussed.