find-0-1-dx-3-e-x-

Question Number 33327 by prof Abdo imad last updated on 14/Apr/18

f i n d \int_{0}^{1} \frac{d x}{3 + e^{- x}}

Commented by math khazana by abdo last updated on 19/Apr/18

l e t p u t I = \int_{0}^{1} \frac{d x}{3 + e^{- x}} c h a n g e m e n t e^{x} = t g i v e

I = \int_{1}^{e} \frac{1}{3 + \frac{1}{t}} \frac{d t}{t} = \int_{1}^{e} \frac{d t}{3 t + 1}

= {[\frac{1}{3} l n (3 t + 1)]}_{1}^{e} = \frac{1}{3} (l n (3 e + 1) - l n (4))

= \frac{1}{3} (l n (3 e + 1) - 2 l n (2)) .