To be honest, we would probably better serve our students in general by presenting integration as a numerical approximation, doing enough symbolic stuff to demo

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.