A change of basis matrix and integrals of power of cosine

This work is based on the idea of Rogers’ (1997) who suggested to use the inverse of a change of basis matrix to find the antiderivative of a function. We use properties of the Chebyshev polynomial of the first kind to derive our change of basis matrix. We give the explicit entries for its inverse and apply it to achieve a formula for integrating the power of cosine.