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

To prove that

\[ \int_0^\infty \lfloor n e^{-x} \rfloor \, dx = \ln \left( \frac{n^n}{n!} \right) \]

where \( n \) is a natural number greater than 1 and \( \lfloor . \rfloor \) denotes the greatest integer function, we will follow these steps:

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