Integral Van Vleck's and Kannappan's functional equations on semigroups

In this paper we study the solutions of the integral Van Vleck's functional equation for the sine integral(S) f(x tau(y) t) d mu(t) -integral(S) f(xyt) d mu(t) = 2f(x) f(y), x, y is an element of S and the integral Kannappan's functional equation integral(S) f(xyt)d mu(t) -integral(S) f( x tau(y) t) d mu(t) = 2f(x) f(y), x, y is an element of S, where S is a semigroup, tau is an involution of S and mu is a measure that is a linear combination of Dirac measures (delta(zi)) i. I, such that for all i is an element of I, z(i) is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d'Alembert's classic functional equation with involution.