A Family of Integrals Related to Values of the Riemann Zeta Function

We propose a relation between values of the Riemann zeta function Formula: see text and a family of integrals. This results in an integral representation for Formula: see text, where p is a positive integer, and an expression of Formula: see text involving one of the above-mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of Formula: see text.