So ...

Upon activation, go to Appearance => Widgets.

You'll notice a new widget-ready area called "Widget for Shortcodes":

Next, you'll have to drag the widget(s) you want to display in your post or page into this new shortcodes sidebar:

And that's all ... ANY widget added to the shortcodes sidebar can be inserted into your post or page content.

All you have to do, is to open your post/page in the post/page editor, and paste the following shortcode (wherever you like in your content):

[do_widget widgetname]

Since I added the calendar widget, my shortcode will look like this ...

[do_widget calendar]

... and here it is pasted into my content:

If the widget name has space in it, then you'll have to add the widget name in quotes. For example:

[do_widget "recent posts"]

OK. Now let's see the result ...