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 \(\zeta(2p + 1)\).