One of my professors used to say that “even a horse can do derivatives. Integration is the real deal”, another one said that you integrate by “look at it, deeply, deeply, deeply; and then solve it”.
The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with equations — once you have it down, you can basically solve it, even if it is harder than the “notice this and that” elegant solution.
Most integration problems require this intuition-based solution which has a certain elegance to it.
It was especially humbling to me that Wolfram alpha fails most of the interesting calculus problems I encountered during my analysis classes, but after a while I managed to solve most of them. But it unfortunately does disappear after not using it for a time..
I actually hit this personally, because right up UNTIL calculus, math was the Feynman method for me. Everything always just clicked, made perfect sense, and I saw the beauty in it, and it was great fun.
Then for calculus, we learned concepts, like what a derivative is, and I understood that, and understood conceptually (as in, what everything "means" and what it tells you) but I could never take that concept knowledge to the practice problems with me.
I could follow along as the professor walked us through a problem, showing us what heuristics helped and what patterns to follow and how to manipulate the functions to get to something that followed one of the patterns to pull an answer out of your ass, but I could never commute those heuristics and patterns to novel examples. It's weird because I was great at doing the exact same thing for physics: Taking a novel and purposely opaque problem and finding which pattern it corresponds to.
To be honest, we would probably better serve our students in general by presenting integration as a numerical approximation, doing enough symbolic stuff to demonstrate that some integrations can be solved that way, doing enough other stuff to demonstrate why there's a ton of integrations that have no closed-form solution with any conventional functions, and then moving on to more productive things rather than blow over a full semester grinding out integrations. Integration by parts is useful as a method for exploring the concept more deeply, and there's a few other such tricks to be used primarily for ensuring the concepts are understood better. But I'm not convinced there's a lot of value in all these integration tricks.
Derivatives are friendly enough that I feel like a certain amount of grinding is justifiable, and it's justifiable as practice for symbolic manipulations in general. But we're leaving a lot of useful stuff on the table while we're jamming down how to integrate with trig identities and other such things.
But it's all pie in the sky anyhow. The Curriculum Must Not Be Changed. The Curriculum Is Perfect. Nothing Can Be Dropped From The Curriculum. I don't know what miracle would have to be worked to get people to reconsider the curriculum from some sensible perspective of what students should be taught rather than the way that question happened to be answered about 100 years ago when the curriculum froze into place, but it probably involves the total destruction of the school system at this point. I can't even get people to process the idea that shoving incomprehensible combinations of 450-year-old words in what is effectively another language at children and telling them this is High True Art is a bad idea, what chance is there of prying away the utterly vital fact that cos(θ/2) = SqRt((1 + cos(θ))/2) out of The Curriculum?
Maybe if colleges continue dropping the SAT and the ACT we can start actually fixing these curricula.
Most integration problems require numerical methods. The ones with with an analytic solution are just where someone at some point found a way of solving them.
The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with equations — once you have it down, you can basically solve it, even if it is harder than the “notice this and that” elegant solution.
Most integration problems require this intuition-based solution which has a certain elegance to it.
It was especially humbling to me that Wolfram alpha fails most of the interesting calculus problems I encountered during my analysis classes, but after a while I managed to solve most of them. But it unfortunately does disappear after not using it for a time..