f-x-1-lnx-1-x-1-1-lim-x-1-f-x-1-2-2-0-1-f-x-dx-

Question Number 95397 by ~blr237~ last updated on 25/May/20

f (x) = \frac{1}{l n x} - \frac{1}{x - 1}

1) \lim_{x \to 1} f (x) = \frac{1}{2}

2) \int_{0}^{1} f (x) d x = γ

Commented by Mikael_786 last updated on 25/May/20

1 . \underset{x \to 1}{l i m} {\frac{1}{l n x} - \frac{1}{x - 1}}

\underset{x \to 1}{l i m} {\frac{1}{x - 1} (\frac{x - 1}{l n x} - 1)} = \underset{x \to 1}{l i m} {\frac{1}{x - 1} (\underset{0}{\int^{1}} x^{y} d y - 1)}

\underset{x \to 1}{l i m} {\underset{0}{\int^{1}} \frac{x^{y}}{x - 1} d y - \underset{0}{\int^{1}} \frac{1}{x - 1} d y}

\underset{x \to 1}{l i m} {\underset{0}{\int^{1}} \frac{x^{y} - 1}{x - 1} d y} = \underset{0}{\int^{1}} {\underset{x \to 1}{l i m} \frac{x^{y} - 1}{x - 1}} d y

= \underset{0}{\int^{1}} y d y = {[\frac{y^{2}}{2}]}_{0}^{1} = \frac{1}{2}

Commented by Mikael_786 last updated on 25/May/20

\underset{0}{\int^{1}} {\frac{1}{l n x} - \frac{1}{x - 1}} d x

l e t l n x = - y \Rightarrow x = e^{- y}

d x = - e^{- y} d y

\underset{\infty}{\int^{0}} {\frac{1}{- y} - \frac{1}{e^{- y} - 1}} (- e^{- y} d y)

\underset{0}{\int^{\infty}} {\frac{e^{- y}}{1 - e^{- y}} - \frac{e^{- y}}{y}} d y = \underset{0}{\int^{\infty}} \frac{e^{- y}}{1 - e^{- y}} d y - \underset{0}{\int^{\infty}} \frac{e^{- y}}{y} d y

\underset{0}{\int^{\infty}} \frac{e^{- y}}{1 - e^{- y}} d y - \underset{0}{\int^{1}} \frac{e^{- y}}{y} d y - \underset{1}{\int^{\infty}} \frac{e^{- y}}{y} d y

l e t z = 1 - e^{- y} \Rightarrow e^{- y} = 1 - z

- y = l n (1 - z) \Rightarrow d y = \frac{d z}{1 - z}

\underset{0}{\int^{1}} \frac{1 - z}{z} * \frac{d z}{1 - z} - \underset{0}{\int^{1}} \frac{e^{- y}}{y} d y - \underset{1}{\int^{\infty}} \frac{e^{- y}}{y} d y

\underset{0}{\int^{1}} \frac{d z}{z} - \underset{0}{\int^{1}} \frac{e^{- y}}{y} d y - \underset{1}{\int^{\infty}} \frac{e^{- y}}{y} d y

\underset{0}{\int^{1}} \frac{1 - e^{- y}}{y} d y - \underset{1}{\int^{\infty}} \frac{e^{- y}}{y} d y = γ

γ : E u l e r C o n s t a n t