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In a German forum, a user asked how the "Feynman"-trick works. The example was $$f(x)=xe^x$$

Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder whether the Risch algorithm is useful for calculating antiderivates by hand. My questions:

  1. Would the Risch algorithm be useful to find antiderivatives of, for example, $xe^x$ by hand?

  2. Does the Risch algorithm always find an antiderivative, if it exists ?

  3. A user of Math Stack Exchange claimed that the fact that the Risch algorithm might not always terminate is not important in practice. Does that mean, that it fails only in "pathological" cases (assuming that it can fail)?

I already asked a question about the Risch algorithm, but I am looking for some details to have an idea of the power and usefulness of the algorithm in cases where the antiderivative can easily be found by integration by parts, substitution or other methods.

And I also would like to know more about the decidability-status of the Risch algorithm.

Pete Caradonna
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Peter
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  • Nice question:). Looking forward for an answer. – MrYouMath Oct 16 '16 at 10:42
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    (https://en.wikipedia.org/wiki/Risch_algorithm) – Jean Marie Oct 16 '16 at 15:36
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    Ad 2: The Risch–Norman algorithm does not always find the antiderivative, even if one exists. The full Risch algorithm, if implemented correctly, does. – Stefan Gruenwald Sep 02 '18 at 15:49
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    Why do these look so similar? https://math.stackexchange.com/questions/4328682/how-to-apply-risch-algorithm-by-hand-to-solve-integrals – mr_e_man Aug 23 '23 at 19:01
  • I would say: no, in general the Risch algorithm won't be carried out by hand. There are certain special cases, however, that can. (These probably go back to Liouville.) For example, Liouvile has a method to compute $\int P(z)e^{Q(z)};dz$ where $P,Q$ are rational functions. Either the method finds the answer in terms of elementary functions, or (more likely) proves there is no answer in terms of elementary functions. https://www.cs.ru.nl/~freek/courses/mfocs-2012/risch/Integration%20in%20Finite%20Terms--Maxwell%20Rosenlicht.pdf – GEdgar Aug 23 '23 at 21:18

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