calculate-0-dx-1-e-x-1-e-x-

Question Number 34267 by math khazana by abdo last updated on 03/May/18

c a l c u l a t e \int_{0}^{+ \infty} \frac{d x}{(1 + e^{x}) (1 + e^{- x})} .

Commented by math khazana by abdo last updated on 07/May/18

c h a n g e m e n t e^{x} = t g i v e

I = \int_{1}^{+ \infty} \frac{1}{(1 + t) (1 + \frac{1}{t})} \frac{d t}{t}

= \int_{1}^{+ \infty} \frac{d t}{(1 + t) (t + 1)} = \int_{1}^{+ \infty} \frac{d t}{{(t + 1)}^{2}}

= {[- \frac{1}{t + 1}]}_{1}^{+ \infty} = \frac{1}{2} .