Prove-that-0-sin-x-e-x-1-dx-1-2-picoth-pi-1-solution-0-e-x-sin-x-1-e-x-dx-0-si

Question Number 169115 by mnjuly1970 last updated on 24/Apr/22

Prove that

Ω = \int_{0}^{\infty} \frac{s i n (x)}{e^{x} - 1} d x \overset{?}{=} \frac{1}{2} (π c o t h (π) - 1)

- - - s o l u t i o n - - -

Ω = \int_{0}^{\infty} \frac{e^{- x} . s i n (x)}{1 - e^{- x}} d x = \int_{0}^{\infty} (s i n (x) \underset{n = 1}{\sum^{\infty}} e^{- n x}) d x

= \underset{n = 1}{\sum^{\infty}} \int_{0}^{\infty} e^{- n x} . s i n (x) d x

= \underset{n = 1}{\sum^{\infty}} \frac{1}{1 + n^{2}} \underset{f u n c t i o n}{\overset{U p s i l o n}{=}} \frac{1}{2} (π c o t h (π) - 1) m . n

N o t e : Υ (s) = \underset{n = 1}{\sum^{\infty}} \frac{1}{s^{2} + n^{2}} = \frac{1}{2 s} (π c o t h (π s) - \frac{1}{2 s})

w h e r e : s \in C - {k i \in Z : k \neq 0}