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I hope this is a good place to ask this as its not a direct math problem.

I've learnt derivatives and integrals for my calculus classes, but I am not overly happy with how these are taught when they become much more complex and it seems to be the case for a lot of others who study calculus at early university.

A lot of the time, we're given a list of "standard integrals/derivatives" that we can then solve what ever is thrown at us.

What we are not taught is how to solve an integral or derivative when you are given some thing that does not resemble any standard one in the list and cannot be manipulated to resemble a standard one given either.

So then how did early mathematicians actually solve them when they had no list of standard ones to refer to - are there methods you can take to try to find the solution and how do you then verify the solution is the correct one?

WDUK
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    Problem solving skill is not something that you can learn, like learning the multiplication table. It's something that you can only practice through solving many problems on your own. In the beginning it is helpful to have a look at how others solve problems, but it wouldn't be any useful to just immitate whatever others do. – WhatsUp Dec 18 '21 at 01:43
  • wolframalpha.com can solve integrals, so if you solve an integral and want to double check, you can see what WolframAlpha says. If it's a definite integral, you can double check with desmos.com (IIRC Desmos can do definite integrals). – joseville Dec 18 '21 at 06:09
  • There used to be a user here who had spectacular integration skills. I vaguely recall that her name was Chloe, maybe. She annoyed people because she just presented answers without explaining how she got them. Can’t seem to find her, now. – bubba Dec 18 '21 at 08:37
  • Aah, found her. It’s Cleo, not Chloe. Take a look at some of her amazing answers. https://math.stackexchange.com/users/97378/cleo – bubba Dec 18 '21 at 08:42
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    @bubba Is there evidence that this "Cleo" is a real person who does the integrations by hand. I can imagine that some machine-aided people created an account and transfer outputs of machines to answers here. This sounds more believable than someone who has a disease that doesn't allow her to write even a brief explanation on the overall method, but somehow can type really lengthy LaTeX formulas correctly and accurately. – WhatsUp Dec 18 '21 at 17:30
  • @WhatsUp. I don’t know anything about Cleo. I prefer to believe that she’s a strange and wonderful genius who does the integrations by hand. No reason for that belief, except that it makes the world a better place. – bubba Dec 18 '21 at 23:08
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    @WhatsUp: The 'disease' is a lie, and this is well-known to all long-time users of Math SE and comes with iron-clad evidence. Unfortunately, most people do not know. In fact, the most likely explanation seems to be that Cleo is a shill account for inverse symbolic expression calculation; a program is used to generate a very large database of many-digit decimal expansions of expressions using well-known primitive functions, and then you simply find your desired number in the list. – user21820 Dec 21 '21 at 10:34
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    @user21820 Thank you, that's exactly what I suspected (: – WhatsUp Dec 21 '21 at 13:23
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    @bubba So she had spectacular integration skills but never showed her work? How do you know she didn’t type the integrals into wolfram alpha? – Radial Arm Saw Jan 17 '22 at 19:47
  • @RadialArmSaw. As I said, I don’t know anything about Cleo. I don’t think she’s just typing things into Wolfram Alpha, because many of the problems she solves are beyond its scope. Maybe she’s using some other software or technology. But, as I said, I just prefer the mysterious genius theory; I don’t really care whether it’s true. – bubba Jan 18 '22 at 00:04
  • @joseville Wolfram alpha does definite integrals as well. – Radial Arm Saw Jan 18 '22 at 00:43
  • @bubba I understand. :) – Radial Arm Saw Jan 18 '22 at 00:43
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    @amWhy. I'm not blind, and I'm not "following" anything or anyone. I prefer the "mysterious genius" theory because it's a better story, and world would be a more interesting place if it were true. Again, I don't really care whether it's actually true or not. Call me a romantic. – bubba Jan 19 '22 at 09:31

1 Answers1

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Differentiation is purely mechanical — you apply one of a few dozen standard formulas, and you’re done.

Integration is different. You’re again given a few dozen standard formulas, but it takes skill to massage the problem into one for which the standard formulas are applicable. There are some standard tricks like partial fraction expansions and the Weierstrass substitution, and others described here, but you still need to be clever and persistent. It takes a lot of practice to be able to recognize the problem patterns and map them to the available solutions. I don’t know of any magic shortcuts, but there’s a pretty long list of semi-magical tricks here.

I spent years getting good at integration, and in some ways I wish I hadn’t. Software like Mathematica and Maple can do the job far better than I can; better than the vast majority of people, actually. I wish I’d spent the time studying Chinese or learning to play guitar, instead. Integration is an amusing intellectual exercise, but I’m not convinced that people need to be good at it, nowadays. I expect your calculus teachers would disagree with me, though, and they’re the ones making the rules that you have to live by.

If you’re trying to find definite integrals in real life (i.e. outside a calculus class), then you’ll probably need to use numerical methods. Many of the functions that arise in science and engineering don’t have nice tidy anti-derivatives.

bubba
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    "I wish I'd spent the time studying Chinese..." So I guess you would agree that learning a language has value, even though we now have translation technology. Similarly, perhaps learning integration has value, even though we now have integration technology. – Dan Dec 18 '21 at 09:34
  • Not an accurate analogy, in my opinion. Translation technology won’t help me much if I’m talking with Chinese friends. Language translation needs to happen in real time, in my brain, and integration doesn’t. – bubba Dec 18 '21 at 10:49
  • It's a valuable analogy, bubba. If you don't get it, that's on you. – amWhy Jan 18 '22 at 20:22
  • We're going to have to agree to disagree, @amWhy – bubba Jan 19 '22 at 09:45