[{quoted}](name=Slushi Simcambi,realm=NA,application-id=3ErqAdtq,discussion-id=Jw8Np7Vx,comment-id=00060000,timestamp=2015-01-12T04:29:00.845+0000)
There is a huge difference to something being core and something being integral to a champion.
Core: the central or most important part of something
Integral: necessary to make whole or complete; essential or fundamental
I don't know about you, but being an A-grade English student, at least through high school, gives me the impression that there is no difference between "Core" and "Integral" items.
The term you are looking for is "Crutch" items.
Vayne gets life steal and damage from botrk, she gets attack speed, she gets the slow. The difference is, she can get that from other items as well. BotRK isn't necessary for Vayne to function as a champion.
You can say that as a person who has quite likely never played Vayne, most definitely not enough to have any comprehension of just how much you're missing if you don't build BotRK on Vayne.
Vayne's kit damage scales pretty evenly between AD (because she's a Marksman, and because of Tumble) and AS (because she's a Marksman, and because of Silver Bolts). An offense item with AD and AS is already going to be valuable.
Then, it also has Lifesteal. Being the only defensive mechanic that Marksmen tend to use (aside from a tiny bit of resistance if they pick up Mercurial or GA), that immediately makes it even more valuable.
The on-hit passive deals %HP damage, very powerfully stacking up with her W. 8% of current HP, plus every 3rd attack dealing 8% of max HP, oh and the weapon has AS, so it's increasing that DPS substantially.
The active %HP and %MS steal gives her a nice chase/escape tool aside from Tumble and Condemn.
You would be hard-pressed to name any reason to not build BotRK, or even to find a reason to build something before BotRK outside of extreme circumstances.
If a champion absolutely needs a certain item for an ability to function the way people think it should in the first place, why shouldn't that ability already provide that function, that way if Riot would want to balance it they could change the length she is invulnerable instead of having to screw with Zhonya's to fix ONE champion.
I guess the primary difference to integral vs. core is one ENHANCES the champion while another FIXES a champion.
Except that's not the case.
Zhonya's doesn't "fix" Morgana. She's perfectly capable of using her ultimate without it. Zhonya's grants her the ability to use it and survive in the middle of the enemy team a lot easier, and it's core/integral to her playstyle, but it's not like she can't use her ultimate when Zhonya's is on cooldown. All Zhonya's does is give her an easier way to handle being in the middle of the enemy team; rather than relying on a defense-heavy build and her Spell Vamp, she can use Zhonya's.
Likewise, BotRK doesn't "fix" Vayne. It just amplifies all of her kit strengths to a degree that is far more worthwhile than any other first items.
You want an example of a crutch?
Mordekaiser is almost literally non-functional without Spell Vamp. His health costs obliterate his own HP, and the shield's constant, instant decay makes it only worthwhile if someone is actively fighting you while you cast spells. You have to have Spell Vamp, whether runes/masteries or in the aforementioned Revolver, for him to work.
Morgana's strength in the middle of an enemy team is amplified by Zhonya's, but it's not impossible for her to work without its effect. It would be a bad idea to not buy Zhonya's, not because she can't do without it, but because it's such a strong choice, you'd be hard-pressed to find an alternate option worth as much. Core/Integral
Mordekaiser's kit is handicapped in a way that requires him to pick up Hextech Revolver or Spell Vamp through runes/masteries. Crutch