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

Question Number 77790 by aliesam last updated on 10/Jan/20

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

Answered by MJS last updated on 10/Jan/20

\int \frac{d x}{\sqrt{- \ln x}} =

[t = \sqrt{- \ln x} \to d x = - 2 x \sqrt{- \ln x} d t]

= - 2 \int e^{- t^{2}} d t = - \sqrt{π} \int \frac{{2 e}^{- t^{2}}}{\sqrt{π}} d t = - \sqrt{π} \erf t =

= - \sqrt{π} \erf \sqrt{- \ln x} + C

\Rightarrow \underset{0}{\int^{1}} \frac{d x}{\sqrt{- \ln x}} = \sqrt{π}

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