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In the topic of integration and anti-derivatives in Calculus we come across cases where the attempt at integration by parts brings us back to the original integral, the most basic example being $\int e^{a x} \cos(b x) \,dx$. Is there a standard name for this type of integrals? I have been calling them "self-referral", but the only usage of this expression that I can find is in health insurance! Have you seen a name in standard textbooks? I have seen "circular integrals" online. What do you call them? Does anyone call it "self-referential"? (Stewart does not have a name for it. Self-referential is an expression in logic. Circular integral might be used for contour integrals.)

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    $\begingroup$ I have never heard of a term for such a thing, nor do I think that they are important enough to bother with a specific name. However, I would avoid the term "circular integral", as I would understand such an integral to either be an integral over a circle, or an integral related to some specific subset of elliptic integrals (not that I know what specific set this would be---that would just be my assumption about the term). $\endgroup$
    – Xander Henderson
    Commented Jan 19 at 18:33
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    $\begingroup$ "self-referential" seems fine. I love how those turn into algebra problems, which is part of what I emphasize. $\endgroup$
    – Sue VanHattum
    Commented Jan 19 at 19:04
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    $\begingroup$ Giving it a name would give the (wrong) impression that it's a feature of the integral itself, whereas it's only a feature of this particular technique we're using to derive a closed form expression for this integral. PS. Justin Skycak's answer provides a name which is free of this issue. $\endgroup$ Commented Jan 19 at 20:11
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    $\begingroup$ Call me a simpleton, but my programming background tells me that you arrive at an equation involving recursion. $\endgroup$
    – TAR86
    Commented Jan 20 at 11:08
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    $\begingroup$ On Mathematics Stack Exchange, it was referred to as "Integration by parts with déjà vu". I found this terminology amusing enough that I use it myself whenever I tutor school students. $\endgroup$
    – Joe
    Commented Jan 21 at 0:23

2 Answers 2

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To my knowledge, this is most commonly known as "cyclic" integration by parts.

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From Does integration by parts with "deja vu" have a name?:

  • Sheard ("Trick or Technique?", 2009) calls it the one-step algebra trick;
  • OP says he saw it called integration by parts with "déjà vu";
  • Glen Wheeler suggests calling it absorption.
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