How to do it ? It’s actually very simple. You can achieve this by unset HISTFILE. Of course in such case we only disable history for current session. What if we want to selectively do not put some commands into bash history ? We can do it via HISTCONTROL variable. From Bash man page:

    A colon-separated list of values controlling how commands are saved on the history list.    
    If the list of values includes ignorespace, lines which begin with a space character are     
    not saved  in the  history  list. A value of ignoredups causes lines matching the previous     
    history entry to not be saved. A value of *ignoreboth* is shorthand for *ignorespace* and     
    *ignoredups*. A value of erasedups causes all previous lines matching the current line to 
    be removed from the history list before that line is saved. Any value not in the above 
    list is ignored. If  HISTCONTROL is unset, or does not include a valid value, all lines 
    read by the shell parser are saved on the history list, subject to the value of HISTIGNORE.
    The second and subsequent lines of a multi-line compound command are not tested, and are 
    added to the history regardless of the value of HISTCONTROL.

In my case I always use export HISTCONTROL=ignoreboth:erasedups and add it to .bashrc if necessary. After that whenever we want to exclude specific command from storing it in the bash history we call it like this <space>mysql -u xxxx -pxxxxx. Remember about space in front of command.

From what I see on Ubuntu 14.04 HISTCONTROL is set to value ignoreboth by default.