Evaluations of some classes of the trigonometric moment integrals

Four classes of the trigonometric moment integrals are evaluated in closed form in a simple and unified manner by making use of the contour integration in conjunction with the Cauchy integral theorem. In all cases, the closed contour of the same shape is used and it is shown that the integrals are expressible only in terms of the Hurwitz zeta function and elementary functions. A number of interesting (known or new) special cases and consequences of the main results are also considered.