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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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    Not exactly what you want but still...http://en.wikipedia.org/wiki/Integration_by_reduction_formulae2011-03-11
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    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!2011-03-11
  • 0
    Others call this "circular integration by parts"...2011-04-12
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    I had a professor who called it the "boomerang method"2011-04-12
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    How do you rearrange the right side so that the integral of f disappears?2015-12-03
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    @RileyE, I love it!2016-06-29
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    @Jon Also sometimes called "loopy" or "looping" integration by parts.2016-07-03

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