  We, by making use of elementary arguments, deduce integral representations of
the Legendre chi function $\chi_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our
earlier established results on the integral representations for the Riemann
zeta function $\zeta(2 n+1)$ and the Dirichlet beta function $\beta(2 n)$ ,$
n\in\mathbb{N}$, are direct consequence of these representations.
