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I am a trivial high school student who loves maths. Recently I found some maths-related YouTube channels. Although they cover a broad range of mathematics, many of them are about integration. So I mainly watched videos about integration.

I found such integrations very difficult or impossible to solve with the scope of the high school maths curriculum. We learn some basic substitution and integration by parts, however, they are insufficient.

For most 'interesting' questions on YouTube, it is not necessary to know the university level of calculus, but the problems require either patience or ingenuity. They often involve several substitutions (which make the calculation very long and complicated) or require creative thoughts that people cannot easily come up with. Challenging questions give me a kind of enthusiasm and encouragement at first, but after 30 mins or so, I feel depressed being unable to solve them and realising my lack of talent.

I fully acknowledge that these YouTubers (or anybody else teaching mathematics and doing such interesting questions) are highly enthusiastic in maths so that they do not hesitate to put a lot of effort learning and practising, and that they are talented and knowledgeable. However, I wonder how they can think so deeply as such, and how they can use such sophisticated arts of techniques to approach to problems.

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    You should by no means feel a lack of talent for being unable to solve such integrals. They are really tricky and require a lot of training. Same thing goes for contest math problems. If you train hard for a few years you would be surprised how much you improve. (Whether the training is the best use of time is a different question.) – littleO Dec 05 '19 at 07:18
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    While nearly all branches of math require patience to learn the tools of the trade, integrals are pretty notorious as being unfun. You either love them or hate them. I tend to fall in the latter camp. I prefer integration theory much more than actually doing integrals. But I can appreciate the artistry of those who do it well. – Rushabh Mehta Dec 05 '19 at 07:29
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    ALSO: note that YouTubers usually aim to wow and impress. That usually means finding the problems with the most gimmicky/one time only solutions. While some YouTubers do try to cover math authentically, it's sad that most portray a type of mathematics that gives the impression that math problems are always solved by divine inspiration. (hint hint: they aren't). – Rushabh Mehta Dec 05 '19 at 07:32
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    Note also that if your goal is to study mathematics in college and become a professional mathematician or if you want to become very good at math competitions, studying these clever integration problems is not going to be helpful. Most professional mathematicians know the same basic techniques of substitution and integration by parts and that plus some mathematical software is totally sufficient for all their integration needs. Contest math often requires clever tricks and insights but rarely of the type of solving a specific non-trivial integral. – quarague Dec 05 '19 at 08:15
  • Some people like to do these integrals. Don Thousand doesn't. Both attitudes are OK. Chacun à son goût. – GEdgar Dec 05 '19 at 12:42

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