advanced-calculus-prove-that-0-1-ln-ln-1-x-dx-ln-1-x-pi-ln-4-

Question Number 137190 by mnjuly1970 last updated on 30/Mar/21

\dots . a d v a n c e d \dots \dots . c a l c u l u s \dots . .

p r o v e t h a t :::

ϕ = \int_{0}^{1} l n (l n (\frac{1}{x})) . \frac{d x}{\sqrt{l n (\frac{1}{x})}} = - \sqrt{π} (γ + l n (4))

Answered by Dwaipayan Shikari last updated on 30/Mar/21

l o g (\frac{1}{x}) = u \Rightarrow x = e^{- u} \Rightarrow 1 = - e^{- u} \frac{d u}{d x}

= - \int_{\infty}^{0} t^{- \frac{1}{2}} e^{- u} l o g (t) d t

= Γ^{'} (\frac{1}{2}) = - \sqrt{π} (γ + l o g (4))

Commented by mnjuly1970 last updated on 30/Mar/21

t h a n k y o u v e r y m u c h \dots