Evaluations of some classes of the trigonometric moment integrals (Q2518292)

The authors evaluate in closed form trigonometric moment integrals of the following type: \[ \int_{0}^{p\pi/q} \phi^{n} \cot \phi \,d \phi,\;\;0<\frac{p}{q}<1,\qquad \int_{0}^{p\pi/q} \phi^{n} \tan \phi \,d \phi,\;\;0<\frac{p}{q}<\frac{1}{2}, \] \[ \int_{0}^{p\pi/q} \phi^{n} \csc \phi \,d \phi,\;\;0<\frac{p}{q}<1,\qquad \int_{0}^{p\pi/q} \phi^{n} \sec \phi \,d \phi,\;\;0<\frac{p}{q}<\frac{1}{2}. \] The results are obtained by applying the contour integration method in conjunction with the Cauchy integral theorem. Several special cases and consequences of the obtained formulae are also considered.