Indefinite integration of oscillatory functions

A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function ${\displaystyle {\int }_x^{y}if\left(t\right)e^{i\omega t}dt}$, -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.