$M_X(t) = \frac{e^{tb} - e^{ta}}{t(b-a)}$

$M_X(t) = \frac{e^{ta} - e^{tb}}{t(b-a)}$

$M_X(t) = \frac{e^{t(b-a)}}{t}$

$M_X(t) = \frac{e^{tb} + e^{ta}}{2}$