Expected value of a time series

I'm reading the fourth edition of Shumway and Stoffer's "Time Series Analysis and Its Applications" and I got stuck trying to determine the expected value or mean function of a simple time series using the definition that is given.

Indeed, Definition 1.1 gives us the mean function of a time series $x_t$ as $\mu_{xt}=E(x_t)=\int_{-\infty}^{\infty}xf_t(x)dx$, where $f_t(x)=\frac{\partial F_t(x)}{\partial x}$ is the density function and $F_t(x)=P\{x_t \leq x\}$ is the distribution function.

Now if, for example, we take $x_t=\beta_1 +\beta_2t$, it makes sense to me intuitively that $E(x_t)=\beta_1+\beta_2t$, but I don't see how to use the definition to get there. How do I determine the density function or the distribution function of a time series like this? Or is that not the point?