0-1-Log-Log-2-x-x-x-5-x-4-x-3-x-2-x-1-dx-Anyone-

Question Number 171665 by Mastermind last updated on 19/Jun/22

Ω = \int_{0}^{1} L o g (\frac{{L o g}^{2} (x)}{x^{x^{5} - x^{4} + x^{3} - x^{2} + x - 1}}) d x

A n y o n e ?

Answered by mindispower last updated on 19/Jun/22

= \int_{0}^{1} 2 l n (- l n (x)) d x - \int_{0}^{1} (x^{5} - x^{4} + x^{3} - x^{2} + x - 1) l n (x) d x

= 2 \int_{0}^{\infty} l n (x) e^{- x} d x + \int_{0}^{1} \frac{1}{x} (\frac{x^{6}}{6} - \frac{x^{5}}{5} + \frac{x^{4}}{4} - \frac{x^{3}}{3} + \frac{x^{2}}{2} - x) d x

= 2 \partial_{a} \int_{0}^{\infty} x^{a - 1} e^{- x} d x ∣_{a = 1} + \frac{1}{36} - \frac{1}{25} + \frac{1}{16} - \frac{1}{9} + \frac{1}{4} - 1

= 2 \partial_{a} Γ (a) ∣_{a = 1} - \frac{5}{6} + \frac{1}{16} - \frac{1}{25}

= 2 Γ^{'} (1) - \frac{5}{6} + \frac{9}{400} = - 2 γ + \frac{- 2000 + 54}{2400}

= - 2 γ - \frac{973}{1200}

Commented by Mastermind last updated on 19/Jun/22

i s t h i s m y q u e s t i o n ?

i f y e s, y o u {d i d}^{'} n t c o m p o s e i t w e l l

Commented by mindispower last updated on 19/Jun/22

l n (- l n (x)) v a l u e o f i n t e g r a l i s c o o r e c t