find-dx-x-e-x-

Question Number 160349 by mr W last updated on 28/Nov/21

f i n d \int \frac{d x}{x + e^{x}} = ?

Answered by mathmax by abdo last updated on 28/Nov/21

\int \frac{dx}{x + e^{x}} = \int \frac{e^{- x}}{1 + {xe}^{- x}} dx = \int e^{- x} \sum_{n = 0}^{\infty} {(- 1)}^{n} x^{n} e^{- nx} dx if ∣ x ∣< 1

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

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

= \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(n + 1)}^{n + 1}} \int t^{n} e^{- t} dt = Γ_{inc} (n + 1) . \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(n + 1)}^{n + 1}}

Commented by mr W last updated on 29/Nov/21

t h a n k s s i r!

Γ_{i n c} (n + 1) s e e m s t o b e d e p e n d e n t f r o m

n . b u t h o w c a n y o u t a k e i t o u t o f t h e

s u m m a t i o n a b o u t n a n d t r e a t i t a s

i n d e p e n d e n t f r o m n ?

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