let-gt-0-find-0-e-x-1-e-2-x-dx-

Question Number 38125 by maxmathsup by imad last updated on 22/Jun/18

l e t α > 0 f i n d

\int_{0}^{\infty} \frac{e^{- α x}}{\sqrt{1 + e^{- 2 α x}}} d x .

Commented by math khazana by abdo last updated on 26/Jun/18

l e t A (α) = \int_{0}^{\infty} \frac{e^{- α x}}{\sqrt{1 + e^{- 2 α x}}} d x c h a n g e m e n t

e^{- α x} = t g i v e - α x = l n (t) \Rightarrow d x = - \frac{1}{α t} a n d

A (α) = - \int_{0}^{1} \frac{t}{\sqrt{1 + t^{2}}} \frac{- 1}{α t} d t

= \frac{1}{α} \int_{0}^{1} \frac{d t}{\sqrt{1 + t^{2}}} = \frac{1}{α} [l n {(t + \sqrt{1 + t^{2}}]}_{0}^{1}

= \frac{1}{α} l n (1 + \sqrt{2}) .

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18

e^{- α x} = t d t = - α e^{- α x} d x

= \frac{- 1}{α} \int_{1}^{0} \frac{d t}{\sqrt{1 + t^{2}}}

= \frac{1}{α} \int_{0}^{1} \frac{d t}{\sqrt{1 + t^{2}}}

u s e f o r m u l a \int \frac{d x}{\sqrt{x^{2} + a^{2}}}