find-I-n-0-1-lnx-n-dx-with-n-fromN-

Question Number 30760 by abdo imad last updated on 25/Feb/18

f i n d I_{n} = \int_{0}^{1} {(l n x)}^{n} d x w i t h n f r o m N

Commented by abdo imad last updated on 27/Feb/18

t h e c h . l n x = - t g i v e I_{n} = - \int_{0}^{+ \infty} {(- t)}^{n} (- e^{- t}) d t

= {(- 1)}^{n} \int_{0}^{\infty} t^{n} e^{- t} d t = {(- 1)}^{n} A_{n} l e t i n t e g r a t e b y p a r t s

A_{n} = {[- t^{n} e^{- t}]}_{0}^{\infty} - \int_{0}^{\infty} - {n t}^{n - 1} e^{- t} d t = n \int_{0}^{\infty} t^{n - 1} e^{- t} d t

= {n A}_{n - 1} \Rightarrow \prod_{k = 1}^{n} A_{k} = \prod_{k = 1}^{n} k . \prod_{k = 1}^{n} A_{k - 1} \Rightarrow

A_{1} . A_{2} \dots . A_{n} = n! A_{0} A_{1} \dots . A_{n - 1} \Rightarrow A_{n} = n! A_{0} = n! \Rightarrow

I_{n} = {(- 1)}^{n} n! .