This is a snippet from the book Introduction to Mathematical Statistics. I am kinda of lost how in this book they calculated the integral $g_{1}(x)$, especially the part where they get $\gamma$($\sum_{1}^{n} x+{\alpha}$).
It would be great, if some one could describe me in steps how this integration reached the solution in (11.1.6) .
Thank you very much
With the change of variable $x=\left(n+\frac 1 \beta\right)\theta$, $$\begin{split} \int_0^{+\infty} \theta^{\left(\sum x_i\right)+\alpha -1}e^{-\left(n+\frac 1 \beta\right)\theta}d\theta &= \frac 1 {\left(n+\frac 1 \beta\right)^{\left(\sum x_i\right)+\alpha}}\int_0^{+\infty}x^{\left(\sum x_i\right)+\alpha -1}e^{-x}dx\\ &=\frac {\Gamma\left(\left(\sum x_i\right)+\alpha\right)} {\left(n+\frac 1 \beta\right)^{\left(\sum x_i\right)+\alpha}} \end{split}$$