How to Solve a Laguerre Integral?

[jije1112]

Homework Statement

this is laguerre integral[/B]

Homework Equations

The Attempt at a Solution

I used 4th equation and 1st

and I stuck here

how to get rid of

or any other idea?
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** hint**solution without

is

 

[Asim Riaz]

The solution without using the Laguerre equations is to use integration by parts. We can rewrite the integral as $$\int_0^{\infty} xe^{-x} dx = \int_0^{\infty} udv $$where $u=x$ and $dv = e^{-x}dx$. Then, $$\int_0^{\infty} xe^{-x} dx = xe^{-x}|_0^{\infty} - \int_0^{\infty} e^{-x}dx$$The first term on the right is $0$. The second term is easily evaluated to be $1$. Therefore, $$\int_0^{\infty} xe^{-x} dx = 1$$