First of all, the idea to "integrate application", no matter what is it, is inherently bad. Libraries should be "integrated" (in fact, just used), not applications. This is not the purpose of most applications.
Also, I would not advise going this way, because this could be either difficult or awkward thing to do (doing just the opposite, using unmanaged code in .NET is much easier).
One approach is using COM components developed in .NET and used in unmanaged code. This is not hard to do, but it's awkward to use the obsolete technology, especially if you realize that the whole .NET technology was developed to get away from such things and introduce purely OOP technology.
Another method is rather exotic but is based on firm CLR standards: direct export of methods to unmanaged. Many would say it's impossible, but that's not true. The number of CodeProject articles demonstrated validity of such approach. Please see my past answers:
loading C# DLL in MFC[
^],
How can I use a dll created in Visual Basic 2008 in Visual Basic 6.0[
^],
Call Managed DLL written in C# from Unmanged Code VC++[
^],
http://www.codeproject.com/Answers/236759/APIs-in-Net-Managed-or-UnManaged-Code#answer2[
^].
—SA