Evaluate the integral $int \ 1/(1+e^x) \ dx$ ?

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Answer 1 · 2017-12-13T14:02:10

$int \ 1/(1+e^x) \ dx = -ln |1+e^(-x)| + C$

We seek:

$I = int \ 1/(1+e^x) \ dx$
$\ \ = int \ 1/(1+e^x) \ e^(-x)/e^(-x) \ dx$
$\ \ = int \ e^(-x)/(1+e^(-x)) \ dx$

Perform the substitution:

$u = 1+e^(-x) => (du)/dx = -e^(-x)$

Then the integral becomes:

$I = int \ 1/(u) (-1) \ du$
$\ \ = - \ int \ 1/(u) \ du$

Which is now a trivial integral, so integrating we get:

$I = -ln |u| + C$

And restoring the substitution:

$I = -ln |1+e^(-x)| + C$

Evaluate the integral int \ 1/(1+e^x) \ dx int \ 1/(1+e^x) \ dx ?