I was reading through this answer on stats.stackexchange, but didn't follow the mathematics behind one step.
They have
$$\int\limits_{\tau=0}^{\infty} e^{-\tau( \text{constant}_1)}\tau^{\text{constant}_2}d\tau$$
and claim that "it a multiple of a Gamma function when integrated over the full range $\tau=0$ to $\tau=\infty$".
Why is that, and how does that help us evaluate the integral?
From the Wikipedia definition of the Gamma Function, I can see that if we just had
$$\int\limits_{\tau=0}^{\infty} e^{-\tau}\tau^{\text{constant}_2}d\tau$$
then this would be equal to
$$\Gamma(\text{constant}_2+1).$$
If $\text{constant}_1$ was summed to the exponent of $e$, then we could just take it out of the integral and we would have the above form. But as it's multiplying an exponent of $e$, I don't know what to do with it.