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Why are there integrals different from Riemann's? That is, what aspect of a function makes it non-Riemann integrable but integrable by other approaches, what are the other approaches, and are their results differentiable or do they not have inverse operations?

George Frank
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    Look up any introduction to Measure theory, it always discusses limitations of Riemann integrals and why people should yuse Lebesgue integrals instead – Alex Dec 29 '13 at 15:32
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    Another alternate is the gauge integral or Henstock-Kurzweil integral http://en.wikipedia.org/wiki/Henstock–Kurzweil_integral – GEdgar Dec 29 '13 at 15:37

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There is an amazing book out there that directly covers most of your question: Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Stieltjes, Henstock-Kurzweil, Wiener and Feynman.

A Garden of Integrals by Frank Burk (published by Mathematical Association of America)

Hope it helps.

Ellie Kesselman
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