You can Intergrate WordPress with CodeIgniter, but it really Depends on what your ultimate application is going to be. But since WordPress and CodeIgniter's file struture is different, you can drop in a codeigniter setup.
The easy way would be like this:
a.) Get Your WordPress Install and Database Working.
b.) Drop in your CodeIgniter Setup, and get the Database Working.
c.) Open the 'index.php' file of the WordPress Core.
d.) Replace it with the CodeIgniter 'index.php' file, but add the following to the bottom of the file right before the codeigniter require:
/** Loads the WordPress Environment and Template */
That will give you a CodeIgniter FrontEnd, with a WordPress BackEnd (administration). You can then go about customizing the CodeIgniter Templates to Interact with The WordPress Setup (posts, pages, etc).
Hope that helped!