0-1-dx-ln-x-by-Gamma-function-

Question Number 97057 by bemath last updated on 06/Jun/20

\underset{0}{\int^{1}} \frac{d x}{\sqrt{- \ln (x)}} ? [b y G amma function]

Answered by Sourav mridha last updated on 06/Jun/20

\boldsymbol l e t \boldsymbol l n (\boldsymbol x) = - \boldsymbol k . .

= \int_{0}^{\infty} \boldsymbol e^{- \boldsymbol k} . \boldsymbol k^{- \frac{1}{2}} \boldsymbol d k

= \boldsymbol Γ (\frac{1}{2}) = \sqrt{π} .

\boldsymbol i t \boldsymbol i s \boldsymbol n e e d \boldsymbol n o t \boldsymbol t o \boldsymbol t e l l -^{″} \boldsymbol b y \boldsymbol G a m m a

\boldsymbol {f u n c t i o n}^{″} .

Commented by bemath last updated on 06/Jun/20

yes . thanks

Answered by mathmax by abdo last updated on 06/Jun/20

I = \int_{0}^{1} \frac{dx}{\sqrt{- lnx}} changement \sqrt{- lnx} = t give - lnx = t^{2} \Rightarrow lnx = - t^{2} \Rightarrow x = e^{- t^{2}}

I = - \int_{0}^{\infty} \frac{- 2 t e^{- t^{2}}}{t} dt = 2 \int_{0}^{\infty} e^{- t^{2}} dt = 2 \times \frac{\sqrt{π}}{2} = \sqrt{π}