I need to find a normalization term $N(\alpha,\beta)$ for the probability density function:
$$PDF(\alpha,\beta)=(x-x_1)^{\alpha}e^{-\beta(x-x_1)}$$
In other words, solve the following equation:
$$N(\alpha,\beta)=\int_{x_1}^{x_2}(x-x_1)^{\alpha}e^{-\beta(x-x_1)}\mathrm{d}x$$
I'm assuming you want some kind of closed form for $N(\alpha,\beta).$
For that, set $\beta(x-x_1)=t.$
Then, $\beta \,\mathrm{d}x=\mathrm{d}t.$
It follows that $$N(\alpha,\beta)=\beta^{-\alpha-1} \int_0^{\beta(x_2-x_1)} t^{\alpha} e^{-t}\, \mathrm{d}t$$
It is known that $$\gamma(s,x)= \int_0^{x} t^{s-1} e^{-t}\, \mathrm{d}t$$
where $\gamma(s,x)$ is the lower incomplete gamma function.
Thus, $$N(\alpha,\beta)=\beta^{-\alpha-1} \, \, \gamma\left[\alpha+1, \beta(x_2-x_1) \right] $$
and we are done.