find-the-value-of-0-t-3-1-e-t-dt-

Question Number 45045 by maxmathsup by imad last updated on 07/Oct/18

f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{t^{3}}{1 + e^{t}} d t .

Answered by maxmathsup by imad last updated on 09/Oct/18

l e t A = \int_{0}^{\infty} \frac{t^{3}}{1 + e^{t}} d t w e h a v e A = \int_{0}^{\infty} \frac{t^{3} e^{- t}}{1 + e^{- t}} d t

= \int_{0}^{\infty} t^{3} e^{- t} (\sum_{n = 0}^{\infty} {(- 1)}^{n} e^{- n t}) d t = \sum_{n = 0}^{\infty} {(- 1)}^{n} \int_{0}^{\infty} t^{3} e^{- (n + 1) t} d t

c h a n g e m e n t (n + 1) t = x g i v e

A = \sum_{n = 0}^{\infty} {(- 1)}^{n} \int_{0}^{\infty} {(\frac{x}{n + 1})}^{3} e^{- x} \frac{d x}{(n + 1)}

= \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(n + 1)}^{4}} \int_{0}^{\infty} x^{3} e^{- x} d t l e t r e m e m b e r t h a t Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t (x > 0) \Rightarrow

\int_{0}^{\infty} x^{3} e^{- x} d x = \int_{0}^{\infty} x^{4 - 1} e^{- x} d x = Γ (4) = (4 - 1)! = 3! = 6 \Rightarrow

A = 6 \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{{(n + 1)}^{4}} = 6 δ (4) w i t h δ (x) = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n^{x}} (x > 0) l e t f i n d δ (x)

i n t e r m s o f ξ (x) = \sum_{n = 1}^{\infty} \frac{1}{n^{x}} w e h a v e δ (x) = - \sum_{n = 1}^{\infty} \frac{1}{{(2 n)}^{x}} + \sum_{n = 0}^{\infty} \frac{1}{{(2 n + 1)}^{x}}

= - 2^{- x} ξ (x) + \sum_{n = 0}^{\infty} \frac{1}{{(2 n + 1)}^{x}} b u t ξ (x) = \sum_{n = 1}^{\infty} \frac{1}{{(2 n)}^{x}} + \sum_{n = 0}^{\infty} \frac{1}{{(2 n + 1)}^{x}}

= 2^{- x} ξ (x) + \sum_{n = 0}^{\infty} \frac{1}{{(2 n + 1)}^{x}} \Rightarrow \sum_{n = 0}^{\infty} \frac{1}{{(2 n + 1)}^{x}} = (1 - 2^{- x}) ξ (x) \Rightarrow

δ (x) = - 2^{- x} ξ (x) + (1 - 2^{- x}) ξ (x) = (1 - 2^{1 - x}) ξ (x) \Rightarrow

δ (4) = (1 - 2^{- 3}) ξ (4) = (1 - \frac{1}{8}) \frac{π^{4}}{90} = \frac{7}{8} \frac{π^{4}}{90} \Rightarrow A = 6 . \frac{7}{8} . \frac{π^{4}}{90} \Rightarrow

= \frac{2.3.7}{2.4} . \frac{π^{4}}{3.30} = \frac{7 π^{4}}{120} \Rightarrow \int_{0}^{\infty} \frac{t^{3}}{1 + e^{t}} d t = \frac{7 π^{4}}{120} ★