Explicit evaluations of the symmetric Euler integral
$\int _{0}^{1}\,{{u}^{\alpha }}{{(1-u)}^{\alpha }}f(u)\,du$
are obtained for some particular functions
$f$
. These evaluations are related to duplication formulae for Appell’s hypergeometric function
${{F}_{1}}$
which give reductions of
${{F}_{1}}(\alpha ,\beta ,\beta ,2\alpha ,y,z)$
in terms of more elementary functions for arbitrary
$\beta $
with
$z=y/(y-1)$
and for
$\beta =\alpha +\frac{1}{2}$
with arbitrary
$y,z$
. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at
$n$
independent randomtimes with uniformdistribution on
$[0,1]$
, then the broken line approximation to the bridge obtained from these
$n+2$
values has a total variation whose mean square is
$n(n+1)/(2n+1)$
.