Prove that `int_0^oo[n e^(-x)]dx=1n((n^n)/(n !)),w h e r en` is a natural number greater than 1 and [.] denotes the greatest integer function..

Explanation

`AAxe[0,oo),n e^(-x)epsilon(0,n]`
If `0ltn e^(-x)1,xepsilon(In n,oo)`.
If `1le n e^(-x)lt 2,xepsilon(In n//2, In n]`
If `2le n e^(-x)lt3,xepsilon(In n//3, In n//2]`
If `n-1le n e^(-x)ltn,xepsilon(0,In n/(n-1)]`
`:. int_(0)^(oo) [n e^(-x)]dx=int_(0)^(In n/(n-1))(n-1)dx+int_(In n/(n-1))^(In n/(n-2))(n-2)dx`
`+............+int_(In n/2)^(In n) 1dx+int_(In n)^(oo)0 dx`
`=(n-1)(In n/(n-1))+(n-2)[In (n/(n-2))`
`-In(n/(n-1))]+..............+1[In n-In-n/2]`
`=In((n^(n))/(n!))`.