  Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma:S\rightarrow S$
is an involutive automorphism. Our main results is that we describe the
solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S}
f(xyt)d\mu(t) +\displaystyle \int_{S} f(\sigma(y)xt)d\mu(t)= 2f(x)g(y),\ x,y\in
S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S}
f(xyt)d\mu(t) -\displaystyle \int_{S} f(\sigma(y)xt)d\mu(t)= 2f(x)g(y),\ x,y\in
S,\] where $\mu$ is a measure that is a linear combination of Dirac measures
$(\delta_{z_i})_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$.
Interesting consequences of these results are presented.
