1-prove-that-0-e-t-ln-t-1-e-t-2-dt-1-2-ln-pi-2-

Question Number 151819 by mnjuly1970 last updated on 23/Aug/21

1 . p r o v e t h a t :

\int_{0}^{\infty} \frac{e^{t} . l n (t)}{{(1 + e^{t})}^{2}} d t = \frac{1}{2} (l n (\frac{π}{2}) - γ)

Answered by Lordose last updated on 23/Aug/21

Ω = \int_{0}^{\infty} \frac{e^{t} \log (t)}{{(1 + e^{t})}^{2}} dt = \int_{0}^{\infty} \frac{e^{- t} \log (t)}{{(1 + e^{- t})}^{2}} dt

Ω (a) = \int_{0}^{\infty} \frac{t^{a} e^{- t}}{{(1 + e^{- t})}^{2}} dt = \underset{n = 1}{\sum^{\infty}} {(- 1)}^{n + 1} n \int_{0}^{\infty} t^{a} e^{- nt} dt

Ω (a) \overset{t = \frac{x}{n}}{=} \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{n^{a}} \int_{0}^{\infty} x^{a + 1 - 1} e^{- x} dx = Γ (a + 1) η (a)

Ω^{'} (a) = Γ (a + 1) (η^{'} (a) + η (a) ψ^{(0)} (a + 1))

Ω = Ω^{'} (0)

ψ^{(0)} (1) = - γ, η (0) = \frac{1}{2}, η^{'} (0) = \frac{1}{2} \log (\frac{π}{2})

Ω = \frac{1}{2} (\log (\frac{π}{2}) - γ)

Commented by mnjuly1970 last updated on 23/Aug/21

b r a v o m r l o r d o s e e x c e l l e n t \dots

g r a t e f u l . .

Commented by Tawa11 last updated on 23/Aug/21

Weldone sir

Commented by Tawa11 last updated on 23/Aug/21

Sir, please prescribe a book I can learn some stuffs like this .

Thanks sir . I just want to be learning this sir .

Commented by Lordose last updated on 23/Aug/21

Advanced calculus explored

Advanced Engineering Mathematics

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