Question

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

Answer 1

`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`