A family of integrals related to values of the Riemann zeta function

We propose a relation between values of the Riemann zeta function $\zeta$ and a family of integrals. This results in an integral representation for $\zeta (2p)$, where $p$ is a positive integer, and an expression of $\zeta (2p+1)$ involving one of the above mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of $\zeta (2p+1)$.