calculate-0-e-nx-dx-

Question Number 87526 by mathmax by abdo last updated on 04/Apr/20

c a l c u l a t e \int_{0}^{\infty} e^{- [n x]} d x

Commented by mathmax by abdo last updated on 05/Apr/20

I_{n} = \int_{0}^{\infty} e^{- [n x]} d x c h a n g e m e n t n x = t g i v e

I_{n} = \frac{1}{n} \int_{0}^{\infty} e^{- [t]} d t = \frac{1}{n} \sum_{k = 0}^{\infty} \int_{k}^{k + 1} e^{- k} d t

= \frac{1}{n} \sum_{k = 0}^{\infty} e^{- k} = \frac{1}{n} \sum_{k = 0}^{\infty} {(e^{- 1})}^{k} = \frac{1}{n} \times \frac{1}{1 - e^{- 1}} = \frac{1}{n} \times \frac{e}{e - 1}

\Rightarrow I_{n} = \frac{e}{n (e - 1)}