I have the probability density function $$f(x) = \frac{1}{\beta} e^{-(x-\theta)/\beta}$$ where $x > \theta$, $\theta \in \mathbb{R}$, $\beta > 0$.
I am trying to find the moment generating function $M_{X}(t)$. I tried \begin{align*} M_{X}(t) &= E(e^{Xt}) \\ &= \frac{1}{\beta} \int_{-\infty}^{\infty} e^{xt} \cdot e^{-(x-\theta)/\beta} \ dx \\ &= \frac{1}{\beta} \left[ \frac{1}{t-\frac{1}{\beta}} e^{xt - \frac{x+\theta}{\beta}}\right] ^{\infty}_{-\infty} \end{align*}
But then I don't really think I can evaluate this, so I'm not sure if I've taken the wrong approach but can't seem to think of any other way. This isn't a homework problem, rather an exam review problem. I know that my desired answer is $\dfrac{e^{t \theta}}{1-\beta t}$, $t < \frac{1}{\beta}$.