x-1-x-1-1-e-t-dt-x-R-prove-that-x-2ln-1-1-e-x-1-1-e-x-1-

Question Number 80612 by M±th+et£s last updated on 04/Feb/20

Ψ (x) = \int_{1}^{x} \frac{1}{\sqrt{1 - e^{t}}} d t \forall x \in R

p r o v e t h a t

Ψ (x) = 2 l n (\frac{1 - \sqrt{1 - e^{x}}}{1 - \sqrt{1 - e}}) - x + 1

Commented by mathmax by abdo last updated on 04/Feb/20

1 - e < 0 s o t h e r e i s a e r r o r i n t h e q u e s t i o n!

Commented by mathmax by abdo last updated on 04/Feb/20

p e r h a p s Φ (x) = \int_{1}^{x} \frac{d t}{\sqrt{1 - e^{- t}}} o r Φ (x) = \int_{1}^{x} \frac{d t}{\sqrt{1 + e^{t}}} o r Φ (x) = \int_{1}^{x} \frac{d t}{\sqrt{1 + e^{- t}}}

Commented by mathmax by abdo last updated on 04/Feb/20

l e t t a k e Φ (x) = \int_{1}^{x} \frac{d t}{\sqrt{1 - e^{- t}}} c h a n g e m e n t \sqrt{1 - e^{- t}} = u g i v e

1 - e^{- t} = u^{2} \Rightarrow e^{- t} = 1 - u^{2} \Rightarrow - t = l n (1 - u^{2}) \Rightarrow t = - l n (1 - u^{2}) \Rightarrow

Φ (x) = \int_{\sqrt{1 - e^{- 1}}}^{\sqrt{1 - e^{- x}}} \frac{2 u}{(1 - u^{2}) u} d u = 2 \int_{\sqrt{1 - e^{- 1}}}^{\sqrt{1 - e^{- x}}} \frac{d u}{(1 - u) (1 + u)}

= \int_{\sqrt{1 - e^{- 1}}}^{\sqrt{1 - e^{- x}}} {\frac{1}{1 + u} + \frac{1}{1 - u}} d u = {[l n ∣ \frac{1 + u}{1 - u} ∣]}_{\sqrt{1 - e^{- 1}}}^{\sqrt{1 - e^{- x}}}

= l n ∣ \frac{1 + \sqrt{1 - e^{- x}}}{1 - \sqrt{1 - e^{- x}}} ∣ - l n ∣ \frac{1 + \sqrt{1 - e^{- 1}}}{1 - \sqrt{1 - e^{- 1}}} ∣ ★