I would like to recommend ["From Calculus to Cohomology"][1]. Contain's several examples and even a chapter 7 : "Applications of de Rham cohomology". It is also so-called "self-contained", but on the downside it does contain some minor flaws which can be quite confusing when reading the material for the first time. I think this is the book you are looking for. 

  [1]: http://www.amazon.com/Calculus-Cohomology-Rham-Characteristic-Classes/dp/0521589568