How to integrate Problem Steps Recorder (psr) in my application?

Free alternative I can suggest is called StepsToReproduce. In many aspects it's much more usable - it allows several options for recording (screen/window/region) and has powerful annotation tools.

It can be used from command line:

psr.exe [/start |/stop][/output ] [/sc (0|1)] [/maxsc ] [/sketch (0|1)] [/slides (0|1)] [/gui (o|1)] [/arcetl (0|1)] [/arcxml (0|1)] [/arcmht (0|1)] [/stopevent ] [/maxlogsize ] [/recordpid ]

Here you can find more information about command line usage: ctrlf5.net/?p=176 link removed as page 176 is now a nasty malware page

And here is sample project which automates psr thorught command-line (c#): http://psr4vs.codeplex.com/SourceControl/changeset/view/10645#172579