dx-e-x-x-

Question Number 64642 by mmkkmm000m last updated on 19/Jul/19

\int (d x) / e^{x} + x

Commented by mathmax by abdo last updated on 20/Jul/19

l e t I = \int \frac{d x}{x + e^{x}} \Rightarrow I = \int \frac{e^{- x}}{{x e}^{- x} + 1} d x i f w e k n o w t h a t ∣ x ∣< 1 w e g e t

I = \int e^{- x} (\sum_{n = 0}^{\infty} {(- 1)}^{n} {({x e}^{- x})}^{n}) = \sum_{n = 0}^{\infty} {(- 1)}^{n} \int x^{n} e^{- (n + 1) x} d x

=_{(n + 1) x = t} \sum_{n = 0}^{\infty} {(- 1)}^{n} \int \frac{t^{n}}{{(n + 1)}^{n}} e^{- t} \frac{d t}{(n + 1)}

= \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(n + 1)}^{n + 1}} \int t^{n} e^{- t} d t = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(n + 1)}^{n + 1}} A_{n}

w i t h A_{n} = \int t^{n} e^{- t} d t t h i s s e q u e n c e c a n b e k n o w n i f w e h a v e

t h e l i m i t s o f i n t e g r a l \dots