How can I run Cygwin Bash Shell from within Emacs?

Question

I am running GNU Emacs on Windows so entering:

M-x shell

launches the Windows command-line DOS shell. However, I would like to instead be a

Answer 1

One more important hint on this subject.

If you use Emacs shell mode and want both bash and cmd sometimes, set it up to use bash by default, because you can type cmd at bash and the resulting dos shell works just fine.

If you setup Emacs to use cmd as the shell (which is the way NT emacs installs), then dos shell works fine, but if you type bash or bash -i to run a bash shell, the resulting shell doesn't work right - see answer 0.

But it works fine if bash is the "first" shell emacs invokes.