Evaluating integral $\displaystyle \int_0^\infty x^n e^\frac{-x}{n} dx$ with Gamma-function?

Question

Consider the following integral:

$\int_0^\infty x^n e^\frac{-x}{n} dx $.

One can find a recursion formula for $R_k = \int_0^\infty x^k e^\frac{-x}{n} dx $:

$R_k = n k R_{k-1}$. This yields $R_n = n^{n+1}n!$

Is there an alternative way of calculating this integral using the Gamma-function $\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx$? The integral just looks like as it could fit into the Gamma-function. However, I don't see the approach (if there is one) ...

Answer 1

Sure. Just do the substitution $x = nt$.