0-1-x-1-x-x-2-2-ln-ln-1-x-dx-3-1-3-ln-6-3-pi-pi-3-27-5ln2pi-6ln-1-6-

Question Number 152142 by Ar Brandon last updated on 26/Aug/21

\int_{0}^{1} \frac{x}{{(1 - x + x^{2})}^{2}} \ln (\ln \frac{1}{x}) d x = - \frac{γ}{3} - \frac{1}{3} \ln \frac{6 \sqrt{3}}{π} + \frac{π \sqrt{3}}{27} (5 \ln 2 π - 6 \ln Γ (\frac{1}{6}))

Answered by mindispower last updated on 28/Aug/21

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

\int_{0}^{\infty} \frac{t^{s} e^{- 2 t}}{{(1 - e^{- t} + e^{- 2 t})}^{2}} d t

f (s) = \int_{0}^{\infty} \frac{t^{s} e^{- 2 t} (1 + 2 e^{- t} + e^{- 2 t})}{{(1 + e^{- 3 t})}^{2}} d t

= \sum_{m ⩾ 0} {(- 1)}^{m} . m \int_{0}^{\infty} t^{s} (e^{- t (2 + 3 m)} + e^{- t (3 + 3 m)} + e^{- t (4 + 3 m)}) d t

Missing \left or extra \right

= \frac{Γ (1 + s)}{3^{s + 1}} \sum_{m ⩾ 1} {(- 1)}^{m} . m (\frac{1}{{(m + \frac{2}{3})}^{s + 1}} + \frac{1}{{(1 + m)}^{s + 1}} + \frac{1}{{(m + \frac{4}{3})}^{s + 1}})

= \frac{Γ (1 + s)}{3^{s + 1}} \sum_{m ⩾ 0} {(- 1)}^{m} (\frac{1}{{(m + \frac{2}{3})}^{s}} + \frac{1}{{(1 + m)}^{s}} + \frac{1}{{(m + \frac{4}{3})}^{s}}

- \frac{2}{3 {(m + \frac{2}{3})}^{s + 1}} - \frac{1}{{(1 + m)}^{s + 1}} - \frac{4}{3 {(m + \frac{4}{3})}^{s + 1}})

= \frac{Γ (1 + s)}{3^{s + 1}} \sum_{m ⩾ 0} (ζ (\frac{1}{3}, s) - ζ (\frac{5}{6}, s) + η (s) - η (1 + s)

+ ζ (\frac{2}{3}, s) - ζ (\frac{7}{6}, s) - \frac{2}{3} (ζ (\frac{1}{3}, s + 1) - ζ (\frac{5}{6}, s + 1))

- \frac{4}{3} (ζ (\frac{2}{3}, 1 + s) - ζ (\frac{7}{6}, 1 + s))

w e w a n t, f^{'} (0)

t o o b e e c o n t i n u e d e t o o m a n y c a l c u l a t i o n

j u s t u s i n g s t e i l j e s c o n s t a n t e

o f l a u r e n t e x p e s i o n z e t a h u r w i t z f o m c t i o n

Commented by Ar Brandon last updated on 28/Aug/21

wow thanks sir