Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral; application to the calculus of ∆-antiderivatives

The theory of dynamic equations appears in the literature in 1988 in the Ph. D. of S. Hilger [18]. The aim of this theory consists in to study differential and difference equations under the same formulation. Thus, the concepts of ∆– derivative and ∆−integral have been defined. These two concepts cover both the classical derivative and the Riemann integral together with the forward and sum operators. After this, a wide number of very important tools have been developed with the intention of solving different kinds of dynamic equations. Functions as the exponential ep(t, s), the trigonometric cosp(t, s) and sinp(t, s) or the hyperbolic coshp(t, s) and sinhp(t, s), together with operators ⊕ and ⊙