Evaluate-0-1-li-2-x-1-x-dx-

Question Number 142791 by mnjuly1970 last updated on 05/Jun/21

E v a l u a t e :: \dots

Ω := \int_{0}^{1} \frac{{l i}_{2} (\sqrt{x})}{1 + \sqrt{x}} d x = ? ?

\dots \dots \dots . .

Answered by Dwaipayan Shikari last updated on 05/Jun/21

2 \int_{0}^{1} \frac{{u L i}_{2} (u)}{1 + u} d u

= 2 \int_{0}^{1} {L i}_{2} (u) d u - 2 \int_{0}^{1} \frac{{L i}_{2} (u)}{1 + u} d u

= 2 \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{2} (n + 1)} - \frac{π^{2}}{3} l o g (2) + \frac{5}{4} ζ (3)

= 2 \underset{n = 1}{\sum^{\infty}} \frac{1}{n^{2}} - \frac{1}{n (n + 1)} - \frac{π^{2}}{3} l o g (2) + \frac{5}{4} ζ (3)

= \frac{π^{2}}{3} l o g (\frac{e}{2}) - 2 + \frac{5}{4} ζ (3)

Commented by mnjuly1970 last updated on 05/Jun/21

t a s h a k o r a n d s e p a s m r p a y a n \dots