How many integration techniques are worth knowing?

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Discussion Overview

The discussion revolves around the necessity and value of learning various integration techniques in the context of classical mechanics coursework. Participants share their experiences with challenging integrals and explore whether mastering these techniques is beneficial for scientific pursuits or if reliance on computational tools is sufficient.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express concern over the difficulty of integrals encountered in coursework, suggesting that multiple substitutions may indicate a lack of familiarity with integration techniques.
  • Others propose that there may be shortcuts or more straightforward methods to solve these integrals, questioning the necessity of extensive substitutions.
  • A few participants argue that reliance on computer algebra systems (CAS) for solving integrals diminishes the need to learn advanced techniques, suggesting that knowing these methods is less critical in the modern context.
  • Conversely, some participants emphasize the importance of understanding integration techniques for verifying results and ensuring accuracy in research and engineering applications.
  • There are differing opinions on whether students should invest time in mastering integration techniques, with some advocating for a focus on computational skills instead.
  • Participants highlight the potential gap in knowledge regarding differential equations and express discomfort with not knowing how to solve them, indicating a broader concern about foundational mathematical skills.

Areas of Agreement / Disagreement

The discussion reveals a lack of consensus on the value of learning integration techniques. While some participants advocate for their importance, others argue that reliance on computational tools makes them less necessary. The debate remains unresolved, with multiple competing views on the topic.

Contextual Notes

Participants express varying levels of familiarity with specific integrals and techniques, indicating that the discussion is influenced by individual educational backgrounds and experiences. There are also references to specific integrals and equations that may not be universally understood without additional context.

  • #31
I would say you need to know all the integration techniques to some degree, basic ones you encounter in calculus 1-3 definitely. This however depends entirely on what your major is, and what you intend to do.
 
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  • #32
Best Pokemon said:
Some guys in this forums found integrals that Mathematica couldn't compute: http://www.wilmott.com/messageview.cfm?catid=10&threadid=85389
That's not hard to come up with. Just write down a horrible contrived expression, take its derivative, and--tada--you got an integral you know the answer to but no one else (human or machine) has any chance of ever calculating.

The question is not whether there are expressions for which a--possibly simple--closed-form analytical expression of its integral exists. Of course there are. The question is: If you do not happen to be able to already know or be able to guess the answer, do you, as a human, have any chance of coming up with an algorithm to get the answer which Mathematica cannot? And this can usually happen only if you have some additional information about the problem (e.g., symmetries, magic coordinate transformation, etc) which Mathematica does not have. Because it sure *DOES* know how to apply all the basic integration techniques, and it does know them much better than you as a human do.

Although I have do admit, I was quite surprised by the BesselJ example. I'll try this some time (maybe some assumptions were missing, and the posted identity doesn't always hold).

btw: I'd bet several of the examples on that page would just work in Mathematica if you turned on the cubic and quartic radicals (which are disabled by default because calculating with them might stress your patience). And the fact that you can just do something like this: Ask the program to solve with techniques you could never hope to apply by hand, is already quite awesome. (actually, most of the symbolic integration techniques one could never apply by hand. Some of them are really cool.)
 
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