Functions that are Khinchin integrable but not Henstock-Kurzweil integrable

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I have read that the Khinchin integral is more general than the Henstock-Kurzweil integral (gauge integral). I am studying gauge integration for a course right now and am not quite at the level to answer this question, but while I'm getting there, I'd like to ask:

What are some Khinchin integrable $f$ which are not Henstock-Kurzweil integrable?

(Aside: this is a sensible question, correct? There are such functions, otherwise nobody would bother to distinguish the Khinchin integral from the less general case of the Henstock-Kurzweil integral, correct?)