Evaluate-0-1-log-x-log-x-1-x-x-1-x-dx-

Question Number 142553 by 777316 last updated on 02/Jun/21

E v a l u a t e :

\int_{0}^{1} \frac{l o g (x) l o g (\frac{x}{1 - x})}{\sqrt{\frac{x}{1 - x}}} d x

Answered by Dwaipayan Shikari last updated on 02/Jun/21

\int_{0}^{1} {l o g}^{2} (x) x^{- 1 / 2} {(1 - x)}^{1 / 2} - l o g (x) l o g (1 - x) x^{- 1 / 2} {(1 - x)}^{1 / 2} d x

= \frac{\partial^{2}}{\partial a^{2}} ∣_{a = \frac{1}{2}} \frac{Γ (a) Γ (\frac{3}{2})}{Γ (a + \frac{3}{2})} - \frac{\partial^{2}}{\partial a \partial b} . ∣_{a = \frac{1}{2} b = \frac{3}{2}} \frac{Γ (a) Γ (b)}{Γ (a + b)}

Answered by mnjuly1970 last updated on 02/Jun/21

s o l u t i o n (g e r c e k b o s s)

Θ (a, b) := \int_{0}^{1} x^{a} {(\frac{x}{1 - x})}^{b - \frac{1}{2}} d x

Θ := \frac{\partial^{2} Θ (a, b)}{\partial a . \partial b} ∣_{a, b = 0}