How to compute $I_n=\int^{+\infty}_{0}x^ne^{-x}dx,\ n \geq 0$?

Question

I want to compute the following integral: $$I_n=\int^{+\infty}_{0}x^ne^{-x}dx,\ n \geq 0$$ How to do it?

Answer 1

You could also define $J(\alpha)=\int_0^{\infty}e^{-\alpha x}dx=\frac{1}{\alpha}$ and then differentiate this expression $n$- times with respect to $\alpha$. We get

$I_n=(-1)^n\frac{d^nJ(\alpha)}{d^n\alpha}|_{\alpha=1}= \frac{n!}{\alpha^{n+1}}|_{\alpha=1}=n!$

Answer 2

Hint

By integration by parts you can find a recursive relation or directly you can use the Gamma function

$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$ and that $$\Gamma(n)=(n-1)!$$