I agree it should. IE totally needs a search box. But Microsoft removed it in all releases starting with IE9. And now with "Spartan", they are killing IE itself.
The only way to have a separate search box in IE is to revert to Internet Explorer 8. You can go to Control Panel -> Programs and Features. Click "View Installed Updates" and then in the Windows Explorer search box, type: Internet. Select the IE version you have installed and click Uninstall. Once you restart and revert to IE8, it will give you the usable search box with drop down menu of search providers.
Using an older version of IE can be problematic because many web pages do not support it. In that case,
see this article on how you can use the Classic Shell Start Menu search box as a replacement for IE's lost search provider functionality. It can give you something like this:
There are addons for IE like Quero toolbar:
http://www.quero.at/news.php?id=82 which give you a toolbar with a search box but it's not very usable.