Could you clarify what you mean by "integrate it into cygwin". Would that mean copying the file into a cygwin directory? Anything else? Or do you just want to link with the library, in which case you don't need to integrate it at all, whatever that means, but just include a reference to the library in the link line, e.g.
gcc myprog.c somelib.a -L/foo/bar -o myprog
This will compile myprog.c, link it with somelib.a and write the myprog executable. -L specifies where the linker should look for libraries, in this case I'm assuming /foo/bar/somelib.a exists.
(You could just do "gcc myprog.c /foo/bar/somelib.a -o myprog" but this gets tedious if you have loads of libraries to include from /foo/bar.)