In today's world of technology and information, truth has become relative to the individual and this truth can change or be distorted based upon environment or circumstance.  Integrity on the other hand is unchanging and always operating from a oneness or wholeness perspective.  
 
Historical Context
Historically truth has been manipulated by those in power to stay in power or increase their power.  History is always told from the perspective of the winners.  Integrity doesn't waver and is steadfast.  It is always doing and saying the same thing.  

Conclusion
For these reasons, perhaps there is some credence for integrity being more important than telling the truth because truth can always be manipulated whereas integrity is always operating from a wholeness concept.