In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic manipulations) and eventually the original integral appears again.

To beginning students, this may superficially appear to be "circular reasoning" that doesn't solve the problem. But it does, because if we have

$\int f(x) dx = g(x) + K \int f(x) dx$

where $K \ne 1$, then rearranging gives

$\int f(x) dx = \frac{1}{1-K} g(x)$.

My question:

Does this technique have a commonly used name? I once saw it called "integration by parts with deja vu" in some supplemental study materials for a calculus course. I don't know who thought of that name but I've taken to using it with my students.

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    $\begingroup$ Not exactly what you want but still...en.wikipedia.org/wiki/Integration_by_reduction_formulae $\endgroup$ – user17762 Mar 11 '11 at 5:25
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    $\begingroup$ Yes, this type of integral is commonly referred to as a "cycler". My prof explained this well and I feel the term really illustrates what's going on here. I won't forget it now! $\endgroup$ – Justin Mar 11 '11 at 5:50
  • $\begingroup$ Others call this "circular integration by parts"... $\endgroup$ – Jon Bannon Apr 12 '11 at 15:42
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    $\begingroup$ I had a professor who called it the "boomerang method" $\endgroup$ – Riley E Apr 12 '11 at 18:57
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    $\begingroup$ @Jon Also sometimes called "loopy" or "looping" integration by parts. $\endgroup$ – Bill Dubuque Jul 3 '16 at 20:43

In research papers I commonly encounter the term absorption for the following step: ($\epsilon \in (0,1)$) $$ f = g + \epsilon f \quad\Rightarrow\quad f = \frac{1}{1-\epsilon}g. $$ In usage one normally sees "absorbing $f$ on the left, we have" or similar. This is very common when estimating norms in the study of partial differential equations.

With integration by parts in particular, I have seen "repeated integration by parts followed by absorption on the left gives". However, as is true of many research level papers, such elementary steps in a proof often go by without any comment at all by the author.


You might be interested in the following paper:

Sheard, M. (2009). Trick or Technique. The College Mathematics Journal, 40, 1, 10-14.

The author explains why it is more a trick than a technique and discusses some applications and extensions. Furhthermore, he calls it the 'one step algebra trick'.


Partial answer from another angle:

From the viewpoint of ring theory, the following is (obviously) true.

Proposition. Let $R$ denote a ring.

Suppose $f,g$ and $\epsilon$ are elements of $R$.

If $\epsilon$ is quasiregular, then

$$f = g + \epsilon f \;\rightarrow\; f = \frac{1}{1 - \epsilon} g.$$

If we can find a name for this proposition, this would amount to a name for the relevant technique.

  • $\begingroup$ Of course, in a field, the quasiregular elements are precisely the elements distinct from $1$. $\endgroup$ – goblin Jun 29 '16 at 15:05

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