[HN Gopher] Learning Feynman's Trick for Integrals
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Learning Feynman's Trick for Integrals
Author : Zen1th
Score : 262 points
Date : 2025-11-29 19:55 UTC (1 days ago)
(HTM) web link (zackyzz.github.io)
(TXT) w3m dump (zackyzz.github.io)
| impossiblefork wrote:
| It starts off with a pretty major error.
|
| I'(t)=\int_0^1 \partial/(\partial t)((x^t - 1)/(ln x))dx =
| \int_0^1 x^t dx=1/(t+1), when it is actually equal to \int_0^1
| x^{t-1}/ln(x)dx.
|
| These two are definitely not always equal to each other.
| owalt wrote:
| No, it is correct. The integral is with respect to x, and the
| ordinary/partial derivatives are with respect to t. Written out
| fully, the derivative computation is
|
| d/dt (x^t - 1)/ln(x) = d/dt [exp(ln(x)t) - 1]/ln(x) =
| ln(x)exp(ln(x)t)/ln(x) = exp(ln(x)t) = x^t.
|
| Edit: d/dt exp(ln(x)t) = ln(x)exp(ln(x)t) by the chain rule,
| while d/dt (1/ln(x)) = 0 since the expression is constant with
| respect to t.
|
| There are convergence considerations that were not discussed in
| the blog post, but the computations seem to be correct.
| impossiblefork wrote:
| Ah, yes. I don't understand how I differentiated with respect
| to x instead of t, but...
| lordnacho wrote:
| 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.
| zeroonetwothree wrote:
| That's how most of math works past high school. It requires a
| lot of practice and intuition.
| lordnacho wrote:
| 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.
| BearOso wrote:
| I think it just tokenizes everything and does pattern
| matching to find compositions it can exploit. It's not
| unlike compiler optimization.
| cyberax wrote:
| 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.
| paulpauper wrote:
| that is fascinating, especially the example integral
| given by henri cohen
| hershkumar wrote:
| "Perhaps I could best describe my experience of doing
| mathematics in terms of entering a dark mansion. One goes
| into the first room, and it's dark, completely dark. One
| stumbles around bumping into the furniture, and gradually,
| you learn where each piece of furniture is, and finally,
| after six months or so, you find the light switch. You turn
| it on, and suddenly, it's all illuminated. You can see
| exactly where you were." - Andrew Wiles
| TrackerFF wrote:
| I see your point, but as it is stated in the article, it is one
| of those techniques that require practice, and time to mature.
| And like it mentions, it's a bit like chess...when you're
| presented with some troubling integral, you can parametrize it
| in a number of ways. Most will bring you back to the beginning
| (like with the standard integration by parts), but the right
| one will make your life much easier.
|
| It can be frustrating when math does not have any clear single
| path, but that's just the nature of the beast. In the beginning
| you'll just have to explore all the paths, but do that a couple
| of hundred times, and you'll start to notice patterns and what
| will work / what will not. Kind of like chess, where a good
| chess player can think N moves ahead in time.
| paulpauper wrote:
| A computer algebra package can try many possible substitutions
| and then show the steps , or help speed up the process for a
| starting point .
| tacitusarc wrote:
| I just finished Mathematica by David Bessis and I wish this
| information was presented in the way he talks about math: using
| words and imagery to explain what is happening, and only using
| the equations to prove the words are true.
|
| I just haven't had to use integral calculus in so many years, I
| don't recall what the symbols mean and I certainly don't care
| about them. That doesn't mean I wouldn't find the problem domain
| interesting, if it was expressed as such. Instead, though, I get
| a strong dose of mathematical formalism disconnected from
| anything I can meaningfully reason about. Too bad.
| chrisweekly wrote:
| That's one of the things I like best about
| https://betterexplained.com -- it focuses on ways to gain
| intuition about a given math concept, using visuals and
| metaphors as appropriate. If only math education were always
| presented like that....
| praptak wrote:
| My intuition for the Feynman's trick is that we construct a
| "morph" which produces the given function (the parameter t
| drives the morphing).
|
| The key to the trick is that we construct the morph so that: a)
| we can tell the rate at which it increases the "area under
| curve" b) the rate is easier to integrate that the original
| function and c) the starting function has a known integral
|
| a) is generally easier because differentiation under integral
| sign lets use use the standard differentiation rules.
|
| b) this is where the difficulty in constructing the morph lies.
|
| So we start from a known value of the integral (from c above)
| and then just add whatever the morph adds, which is the
| integral of the rate from a) over the interval of the morph.
| zeroonetwothree wrote:
| It's interesting he mentions he doesn't like contour integration
| since many integrals can be done either way.
|
| Feynman's trick is equivalent to extending it into a double
| integral and then switching the order of integration.
| measurablefunc wrote:
| Don't forget to check for the necessary measurability &
| integrability of the sections (f(a, y), f(x, b)) before
| switching the order: https://en.wikipedia.org/wiki/Fubini%27s_t
| heorem?useskin=vec....
| dunham wrote:
| This reminds me of the "snake oil method" for generating
| functions. It's been many years, but I remember it as adding
| another sigma and then swapping them.
| biophysboy wrote:
| > So I got a great reputation for doing integrals, only because
| my box of tools was different from everybody else's
|
| This is the most important lesson I learned in grad school.
| Methods are so important. I really think it is the core of what
| we call "critical thinking" - knowing _how_ facts are made.
| analog31 wrote:
| 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 "form" of
| a problem and applying an algorithm appropriate for that form,
| drawn from the most recent chapter.
|
| In the TFA, the natural log in the integrand was a dead give-away
| because it only comes from one place in the standard order of
| topics in calculus class.
|
| Is this what we call intuition?
|
| The students called this the "trick." Many of them had come from
| high school math under the impression that math was subjective,
| and was a matter of guessing the teacher's preferred trick from
| among the many possible.
|
| For instance, all of the class problems involving maxima and
| minima involved a quadratic equation, since it was the only form
| with an extremum that the students had learned. Every min/max
| problem culminated with completing the square. I taught my
| students a formula that they could just memorize.
|
| The whole affair left me with a bad taste in my mouth.
| charcircuit wrote:
| 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.
| eru wrote:
| Depends on your sense of fairness. Math Olympiads don't test
| what's in the book, but they are also fair.
| charcircuit wrote:
| The fairness of a competition and the fairness of asking a
| particular question are different.
|
| 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.
| Enginerrrd wrote:
| I actually think math and sciences should introduce what I
| call "synthesis" much earlier. i.e. I don't think it's unfair
| to give students all the ingredients and add in a question on
| the exam to see if they can take those ingredients and apply
| it to a problem type they haven't seen before. (This is a
| great differentiator between C students and A students.) Or
| for a science class, rather than perform an experiment, I
| think the students should have to actually DESIGN the
| experiment first. (I had one laboratory exam in 2nd semester
| undergrad chem class that did this and it was amazing! The
| students also performed pretty well at it too. It consisted
| of being told to figure out how much zinc was in a lozenge.
| We were also maybe given a handy reaction formula and that
| was it. You had to design your analysis procedure and figure
| out how to get the quantity you wanted out of it, and then
| actually perform your analysis all within the exam period.)
|
| 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.
| zamadatix wrote:
| I think the difference is something like Feynman's trick
| simplifies a hard integral by introducing a parameter and
| differentiating the whole integral, while substitution
| simplifies an integral by changing variables to undo the chain
| rule. But it has been so long since I've done integration
| manually I'm not 100% sure that's an accurate description/the
| full story.
|
| The thing I hated about integration was which approach would
| work and the best option for each approach were much more "do a
| lot and see what's right" and I was too lazy :).
| jjtheblunt wrote:
| https://en.wikipedia.org/wiki/Risch_algorithm
|
| is super interesting, related to your last sentence.
| jmyeet wrote:
| Back in college I stopped doing maths in second year as a major
| because of the way it was taught. I just hated it. Numerical
| methodds in particular broke me. My main problem was we never
| really got told how things fit together. Resources like
| 3blue1brown just didn't exist at that time, sadly. We just had
| dusty and expensive and very dry textbooks to rely on. For
| example, we just got through into ODEs and were told "just use
| e^at". We started doing contour integrals without really telling
| us what was going on. Honestly, things like linearity were never
| really taught for basic stuff like derivatives and integrals.
|
| But I had always loved maths and went back to it much later.
| After having done some computer science, some concepts just made
| it click more for me. Like sets were a big one. Seeing functions
| as just a mapping between sets. Seeing functions as set elements.
| Seeing derivatives and integrals as simply the mapping between
| sets of functions.
|
| What fascinates me is that differentiation is solved, basically.
| Don't come at me about known closed form expressions. But
| integration is not. Now this makes a certain amount of sense.
| Differentiation is non-injective after all. But what's more
| fascinating (and possibly really good evidence of my own
| neurodivergence) is that integration isn't just an algorithm. It
| requires some techniques to find, of which the Feynman technique
| is just one. I think I was introduced to it with the Basel
| problem. I have to confess I end up watching daily Tiktok
| integration problems. It scratches an itch.
|
| I kinda wish I'd made it to complex analysis at least in college.
| I mean I kinda did. I do remember doing something with contour
| integrals. But it just wasn't structured well. By that I mean
| Laplace transforms, poles of a function in the S-plane and
| analytic continuations.
|
| I'm not particularly proficient at the Feynman technique. Like I
| can't generally spot the alpha substitution that should be made.
| Maybe one day.
| CyberDildonics wrote:
| This seems like a bizarre comment that has almost nothing to do
| with the title.
| pvitz wrote:
| When I was a student of physics and came across this paragraph in
| Feynman's book, I was curious if he really meant the simple
| technique explained in the article, a more general one (also
| described in the article with the integral bounds as functions of
| a parameter) or something else. I don't know, but this led me to
| read the text "Advanced Calculus" by Edwin Bidwell Wilson (1912),
| which includes a lot of examples and gems. If there is some young
| student out there who wants to go well beyond the basic
| techniques of calculus taught in analysis or mathematical physics
| courses, have a look at [0].
|
| [0]
| https://archive.org/details/advancedcalculus031579mbp/mode/1...
| kqr wrote:
| 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?
| EGreg wrote:
| Frustration. A lot of frustration:
| https://en.wikipedia.org/wiki/Path_integral_formulation
| ErroneousBosh wrote:
| I spend a lot of time working with real-world electronics,
| where a good mathematical background is important to calculate
| things like component values for a desired behaviour.
|
| 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 20.7kO
| and 21.3kO 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 22kO resistor and an
| 18kO resistor in series with a 4.7kO pot, because that is going
| to need adjusted on test because of the tolerances in
| everything else", wouldn't we?
| agarttha wrote:
| In quantum mechanics, what you can measure experimentally
| (observables) are given by integrals. You can do the integrals
| computationally, but then you only have an empirical
| understanding of how the observables behave when you change
| some parameter of your experiment.
|
| 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.
| srean wrote:
| Often times it so happens that the point of interest is not the
| numerical value of the integral but its behavior at different
| points in its domain. If I am able to figure out the expression
| it becomes easier.
|
| 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.
| FabHK wrote:
| Extraordinarily well done didactically, by the way:
|
| 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.
| paulpauper wrote:
| Feynman's Trick is one of those things where it sounds great in
| theory, but then it's hard to know where it works or how to apply
| it when the example has not already been chosen in advance to
| exploit it.
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