To evaluate
\int \frac{1}{e^x + 1} dx
First make the substitution u=e^x +1 and work through to obtain
\int \frac{1}{u^2 -u} du
Next http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/GeomCS.html on the bottom line to obtain.
\int \frac{1}{(u-\frac{1}{2})^2 -\frac{1}{4}} du
Now let y=u-\frac{1}{2} to obtain
\int \frac{1}{y^2 -(\frac{1}{2})^2} dy
This is a known integral, you should have it in your log tables and eventually with practice will become somewhat familiar with it. This integral is
\int \frac{1}{y^2 -a^2} dy=\frac{1}{2a}\log\left(\frac{y-a}{y+a}\right)
So here a=1/2 and the integral is
\log\left(\frac{y-\frac{1}{2}}{y+\frac{1}{2}}\right)
Putting back u
\log\left(\frac{u-1}{u}\right)
Finally put back x
\log\left(\frac{e^x}{e^x +1}\right)
Which you can check is the right answer. You should check the differenciation to get a feel for how the derivative works here.
The key to this question was knowing how to complete the square for the inverse polynomial part and of course being willing to make a second substitution.
