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To people who find this stuff useful in practise today (and not merely fascinating or useful 50 years ago): what is your line of work? I have needed to know the values of a few integrals in my job, but I have always ended up with a close enough answer using computational methods. What am I missing by not solving analytically?
Earlier quoted context omitted.
I think it's intuitive to assume what you are being tested on is what is being taught by the book or the teacher. It's unfair otherwise.
Depends on your sense of fairness. Math Olympiads don't test what's in the book, but they are also fair.
When things are just for fun, the impact of having unfair questions than when unfair questions can cause people to fail a class or get a lower GPA. This is why sometimes these kind of unfair questions get designated as extra credit as its unfair for these questions to actually count against you.
To people who find this stuff useful in practise today (and not merely fascinating or useful 50 years ago): what is your line of work? I have needed to know the values of a few integrals in my job, but I have always ended up with a close enough answer using computational methods. What am I missing by not solving analytically?
But far better is developing a sense of what's "about right".
I have taught people who studied Electronic Engineering "properly" who calculate that the resistors need to be 20.7kΩ and 21.3kΩ for a given circuit and then will go mad scouring Farnell, Mouser et al for those values.
You or I would say "That needs to be a 22kΩ resistor and an 18kΩ resistor in series with a 4.7kΩ pot, because that is going to need adjusted on test because of the tolerances in everything else", wouldn't we?
To people who find this stuff useful in practise today (and not merely fascinating or useful 50 years ago): what is your line of work? I have needed to know the values of a few integrals in my job, but I have always ended up with a close enough answer using computational methods. What am I missing by not solving analytically?
In our experiments, we need to know how the frequency of an electromagnetic resonator will change when we couple it to a quantum system. We calculate these frequency shifts with integrals. Being able to calculate these integrals analytically for some limiting cases helps us understand the dependence on the parameters. And usually you can patch the limiting cases together and not even have to compute the integrals numerically.
First, a motivational anecdote, then some straightforward theory, a simple (yet impressive) example fully worked out, the general method, and further examples of increasing difficulty for practice with hints.
To people who find this stuff useful in practise today (and not merely fascinating or useful 50 years ago): what is your line of work? I have needed to know the values of a few integrals in my job, but I have always ended up with a close enough answer using computational methods. What am I missing by not solving analytically?
To give an example consider the moment generating transform, Laplace transform. Their symbolic expression can be very informative.
Consider the Mercator projection. It was designed without any idea of the closed form of the required integral. It was mostly done by estimate and gut feel. Now that we know the actual form (an entirely serendipitous discovery) we feel more confident that we understand the transform. This part is considerably psychological but not entirely.
Note that when drawing a map in Mercator projection we have to fall back to numerical estimation. But it helps that parts of the transforms are built from functions tha have names, that means we have seen the same functions elsewhere, it instills a sense of familiarity and understanding.
There are way to many functions to name, so the ones we have given names to are a bit special.
I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "for…
I think it's intuitive to assume what you are being tested on is what is being taught by the book or the teacher. It's unfair otherwise.
I think not doing this starting in like middle school is a big part of the reason why people think math/science is useless. Unless the exact scenario they have been taught pops up, they can very rarely see the application. But the real world NEVER works this way. A problem is NEVER formulated as a straight forward well-formed problem. Figuring out how to mold it into something that you can apply the tools you know to is in and of itself a REALLY important skill to practice, and sadly, we almost NEVER practice that. Only in grad school does that type of thing come up.
Earlier quoted context omitted.
I don't know about this particular case though, I get the feeling there's a system to it that can be exploited by eg Wolfram. It's just that you're in the dark for a long time before you find the switch.
Your intuition is right. There is a general algorithm for finding the antiderivatives: https://en.wikipedia.org/wiki/Risch_algorithm Its simplified form can solve pretty much all the undergrad antiderivation problems. I'm a math major, but I consider the time spent learning the tricks for antiderivation to be kinda useless.
My issue with both this and u-substitution is that you don't know what expression to use. There are a LOT of expressions that plausibly simplify the integral. But you have to do a bunch of algebra for each one (and not screw it up!), without really knowing whether it actually helps. OTOH, if I'm given the expression, it's just mechanical and unrewarding.