I need to verify if an MLE is biased for this Gamma statistical model.
\begin{align*} \mathbb{E}\left[\frac{1}{\bar{X}}\right]&=\int^\infty_0\frac{1}{s}\frac{(n\beta)^{2n}}{\Gamma(2n)}s^{(2n-1)}e^{-n\beta s}ds\\ &=\frac{(n\beta)^{2n}}{\Gamma(2n)}\int^\infty_0s^{(2n-1)-1}e^{-n\beta s}ds\\ &=\frac{(n\beta)^{2n}}{\Gamma(2n)}\int^\infty_0s^{(2n-1)-1}e^{-n\beta s}\color{red}{\frac{(n\beta)^{2n-1}}{\Gamma(2n-1)}\frac{\Gamma(2n-1)}{(n\beta)^{2n-1}}}ds\\ &=\frac{(n\beta)^{2n}}{\Gamma(2n)}\frac{\Gamma(2n-1)}{(n\beta)^{2n-1}}\\ &=\frac{n\beta}{2n-1}\\ \end{align*}
Any hints on how the red part was derived? I got up to that part, but needed to look up the solution.
My assumption is that because that some constant multiplied by the integrand on the second line is equal to 1. Using using the form of the Gamma distribution, we can find that constant, the bit in red is just 1 written in a fancy way?