What is the adjectival form for the word "integrity?"

I don't know of an adjective form of integrity, so I can't solve your issue directly. However, depending on if your use case allows it, you might consider the phrase Have integrity. Obviously if you're trying to offer a set of parallel Be X statements, this won't work, but if you're looking for a concise two-word imperative sentence, I think it carries the meaning you're looking for.

Looking in some dictionaries, it seems there is no such word. There is integer in German, though, and íntegro in Spanish, both of which have the intended meaning. English translations that come up are upright and honourable, but nothing like integer or integral or anything like that.

If you don't like what you have so far (Yahoo Answers' "there is no noun [sic!] with the same root, use 'upright'", or the many suggestions in that WordReference thread — "a good sort", "decent bloke", "a man of integrity", "a man of good character", "principled", "reasonable" and whatnot), then you can use this:

Be a mensch.

Merriam-Webster defines mensch thusly:

a person of integrity and honor

Wiktionary even has a few cites:

• 1960, The Apartment:
  Doctor Dreyfuss [to C. C. Baxter]: Be a mensch!
  
  
• 2005, Alan Hollinghurst, The Line of Beauty, Bloomsbury Publishing, page 428:
  Lionel Kessler, relaxing perhaps on a Louis Quinze day bed, garlanded all round with lines of beauty, seeing welcome proof that his clever maligned young friend was a mensch.
  
  
• 2008 December 28, George Solomon, “My Little Red Book”, The Washington Post, page D01:
  Olie Kolzig: Goalie for the Washington Capitals who spent most of 16 seasons between the pipes for the team until being released in 2008. Had the longest career of any Capital. Now plays for Tampa Bay. The ultimate mensch, in my book.

Edit two years later in reply to comments. No, this word does not have to be spoken to someone with a certain cultural baggage. Here is what Steve Martin tweeted earlier today:

I could not be more stunned by the loss of Robin Williams, mensch, great talent, acting partner, genuine soul.

This went out to 5.2 million followers, not a homogenous bunch by any measure, then got retweeted 50000 times, then picked up by traditional media outlets all over the world.

Mr. Martin does not exactly have a track record of not knowing when to use which word, and we cannot with a straight face claim that everyone who read his message shares the same cultural baggage.

Even more to the point, even if every single recipient did have to look up any word in that message because it was somehow obscure — now it no longer is.

Virtuous: Having or showing virtue, especially moral excellence: led a virtuous life.

Note that integrity is the quality of being honest and having strong moral principles.

Be virtuous!

Integrous

http://en.wiktionary.org/wiki/integrous

http://www.allwords.com/word-integrous.html

It doesn't pass a spell-checker, but there are over 500,000 hits for it on Google, most of which are using it precisely as you defined.

Why limit the word to a form of "integrity" which from what I've seen is a bit sterile, when there are so many other words that convey the meaning so beautifully:

Be noble

Be upstanding

Be moral

Be without reproach

integrious Variations of usage found in literature include integrous and integritous, also with similar meaning.

Wiktionary:

integrious (comparative more integrious, superlative most integrious)
(rare) Marked by integrity.
Howard is an integrious man because his values are congruent with and evident in his words, actions, personality and life.

Most importantly, Wiktionary cites the reference of
The Oxford English Dictionary 2nd Edition Volume VII.

Considering the marking "rare" along with the OED citation, it appears that it is (or was) an authentic word.

Found something on the Why We Need the word Integrious movement!

While the use of the word "integrous" or "integrious" (variations of an adjective form of "integrity") is commonly used in spoken American English, especially in educational and professional forums, it is not commonly used in written American English - hence the lack of inclusion in dictionaries.

To directly answer your question, there is really no adjective for integrity. However, there are several that could capture the essence of the word, although not in its entirety.

Thus, a few examples are:

• Be honest
• Be true to yourself
• Be upright
• Be blameless
• Be above reproach

Few, if any, however, are colloquial. My suggestion would be to select the word(s) (and there are many) that represent the aspect of integrity you deem most important for the occasion.

Be Ethical

Ethical - of or relating to moral principles or the branch of knowledge dealing with these.

Integrity - the quality of being honest and having strong moral principles.

Depending on whom I was writing to or for, I might be brave enough to try 'intact'. 'John was a thoroughly intact individual, so I decided to trust the job to him'. The Latin 'intacta' means 'unimpaired'. The second meaning of 'integrity' in the ODE (not OED) is 'the state of being whole and undivided'. So my contention is that an honest person is an 'intact' person.

But then it would be just as quick to say 'As John was a man of integrity, I decided to trust the job to him'.

I use "integrity" with information a good deal. If information has integrity, it is "reliable" and/or "valid." For a person, I'd probably refer to them as "reliable" if they display integrity.

Integrated.

Not really the correct connotation when describing a person, but yeah. That's the adjective form of the noun "integrity."

Describing a person, I would favour 'virtuous'. In the context of data integrity, though, I have heard industry people use 'integral' to describe data.

I would go with trustworthy, meaning "able to be relied upon as a person of integrity".