\begin{align*} M(t) & = \int_0^{\infty} e^{tx} \frac{1}{\mu} e^{-\frac{x}{\mu}} dx\\ & = \frac{1}{\mu} \int_0^{\infty} e^{- x \left ( -t + \frac{1}{\mu} \right ) } dx\\ & = \frac{1}{\mu \left(-t + \frac{1}{\mu} \right )} e^{- x \left ( -t + \frac{1}{\mu} \right ) } \big |_0^{\infty}\\ & = \frac{1}{\mu \left(-t + \frac{1}{\mu} \right )} \left ( -0 + 1 \right )\\ & = \frac{1}{\left(-\mu t + 1 \right )}. \end{align*}