There are a ton of ways to integrate Cygwin with Emacs on Windows. EmacsWiki shows a few ideas. Here are the options that I\'ve found:
Build emacs from source:
-Download the latest stable release (24.5 as of this writing).
-Unzip and cd into the emacs folder
-Run these steps in order:
$ ./autogen.sh
$ ./configure --with-w32
$ make
$ make install
-Copy emacs.exe, by hand, from the "emacs folder"/src to cygwin/bin
-Run emacs:
$ emacs&