Looking for an accessible explanation of Henstock–Kurzweil (gauge) integral I'm not completely new to analysis, but I'm an engineer -- very applied, not very theoretical -- looking into self-studying pure mathematics. I've recently stumbled upon Henstock–Kurzweil integrals; many internet articles state that it's a very intuitive idea, but I simply can't wrap my head around its intuitive meaning as successfully as I can do it for the Riemann integral. The Wikipedia article seems to be pretty well-written, but I probably need a simpler stated approach/definition, because I'm only starting to get into all of these things.
Can someone give me their own explanation of what the Henstock-Kurzweil is, or perhaps a good resource?
 A: There is a nice beginner's treatment in section 8.1 of Abbott's Understanding analysis.
Abbott's main source is Bartle's article "Return to the Riemann integral."  JSTOR link, full article
Abbott also recommends The integral: an easy approach after Kurzweil and Henstock by Lee and Výborný and A modern theory of integration by Bartle for more detailed treatments.
A: In my opinion the first and foremost place to explore should be Eric Schechter's webpage dedicated to the so-called gauge integral (linked on the WP page you mention). 
One should probably also mention the book The Integral: An Easy Approach after Kurzweil and Henstock by Lee Peng Yee and Rudolf Vyborny, and, if you happen to read French, two very accessible introductory texts due to Jean-Pierre Demailly and available as [E3] on his webpage of teaching documents (their bibliography includes Yee and Vyborny's book).
A: How about the Carus Monograph:
R. M. McLeod, Generalized Riemann Integral (Carus Mathematical Monographs) (Mathematical Association of America, 1982)
It may be out of print, so look in libraries.
