Four Classes of Definite Integrals about Hyperbolic and Trigonometric Functions

Abstract

By establishing recurrence relations and then determining boundary values, we examine four classes of definite integrals of xm over higher powers of cosh x, sinh x, cos x and sin x in denominators. They are explicitly evaluated in terms of the logarithm function, the Riemann zeta function and its variants, such as Dirichlet beta function and Legendre’s chi-function.