Representation of a trigonometric integral via combinatorial sums

In [Appl.Math.E-Notes 22 (2022) 637-643], Stewart expressed a series involving the Catalan numbers in terms of a trigonometric integral I(k) (k = 0, 1, 2, . . .).He asked whether there exists a general expression for this integral, which shows that I(k) is rational for each k.Here, we study four combinatorial sums and show that each of these sums is equal to I(k).In particular, we give an affirmative answer to Stewart's question and settle a conjecture attributed to Sangchul Lee.