SQL Server 2005/2008 logging connections How can I audit SQL connections to the database from  


Windows credential
Sql Users


Is there some built-in feature where I can determine number of:


all connection
active connection
unclosed connection
current status
historical prevue


I hope that I'm going right way. I have application that takes all RAM to the SQL Engine and transaction logs raising to the unbelievable size on HDD durring one-two days.
No, you're not going about this in the right way.


  I have application that be able to take all RAM to the SQL Engine


SQL Server will use as much as it needs unless you limit it. Read Brent Ozar's A Sysadmin’s Guide to Microsoft SQL Server Memory.


  transaction logs raising to the unbelievable size on HDD during
  one-two days


9 times out of 10 this is due to the database being in FULL recovery and no log backups being taken. See Paul Randall's article Running Out of Transaction Log Space.

If you want to monitor the items you listed:


Using SQL Server Profiler
sp_whoisactive
SQL Server Perfmon Best Practices