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

1 Answer
Dec 13, 2017

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

Explanation:

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 #