Add different applications to MDI

First of all, the question makes no sense. There is no such thing as "add application to an application".

If you simply mean that you want to run an application (start a child process), you can do it, but what are you going to do with it? Processes are very well isolated. If they are not intentionally designed for some collaboration, you cannot do much about it. Trying to do such things is not recommended. Even though you can insert a main window of child application (when and if it appears) in some of your forms, this is a dirty trick which will makes the situation even worse. It works with Windows only because its legacy functionality inherited from old Windows versions when Windows was not an OS yet. If will run inside some host form, with all its non-client areas, which, in turn, will brake all the acceptable UI styles; and you still won't have a real control over the application. Don't even think about it.

However, if the child application is a console-only application, you can pretty easily redirect StandardOutput and show the results in your form. The idea is apparent from this MDSN help page, with a code sample:
http://msdn.microsoft.com/en-us/library/system.diagnostics.process.standardoutput.aspx[^].

Now, about your MDI…

Here is the idea: who needs MDI, ever? Why torturing yourself and scaring off your users?
Do yourself a great favor: do not use MDI at all. You can do much easier to implement design without it, with much better quality. MDI is highly discouraged even by Microsoft, in fact, Microsoft dropped it out of WPF and will hardly support it. More importantly, you will scare off all your users if you use MDI. Just don't. Please see:
http://en.wikipedia.org/wiki/Multiple_document_interface#Disadvantages[^],
How to Create MDI Parent Window in WPF?[^].

I can explain what to do instead. Please see my past answers:
How to Create MDI Parent Window in WPF? [Solution 2],
Question on using MDI windows in WPF[^],
MDIContainer giving error[^],
How to set child forms maximized, last childform minimized[^].

—SA