Feynman's Integral Trick with Math With Bad Drawings

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Richard Feynman famously used differentiation under the integral sign to solve many difficult problems, including one during his time at Los Alamos Laboratory during World War II that had stumped researchers for 3 months.
Learn how Feynman's Integral Technique works from Oxford Mathematician Dr Tom Crawford with the help of Ben Orlin from Math With Bad Drawings and his brilliant cartoons.
You can find out more about Ben at mathwithbaddrawings.com/
Get Ben's latest book 'Change is the Only Constant' from Amazon here: www.amazon.co.uk/Change-Only-...
Produced by Dr Tom Crawford at the University of Oxford.
Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
For more maths content check out Tom's website tomrocksmaths.com/
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
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3 июл 2024

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Комментарии : 263

@TomRocksMaths 3 года назад

Ben's cartoons are excellent, but don't just take my word for it, check them out for yourself here: mathwithbaddrawings.com/

@TheBigBawsss 3 года назад

Never expected machine gun kelly to explain Feynman integration technique, but here we are

@TomRocksMaths 3 года назад

I saved pop punk, so now its time to save maths...

@zzzzzzzjsjyue2175 2 года назад

@@TomRocksMaths ratio

@Joshua-th5sg Год назад

@@zzzzzzzjsjyue2175 insanely massive L

@quidlad2050 Год назад

@@zzzzzzzjsjyue2175 YOOOOOO SICK RATIO

@The-Devils-Advocate Год назад

@@zzzzzzzjsjyue2175 lol

@PapaFlammy69 3 года назад

Gud moanin' :3

@TomRocksMaths 3 года назад

Well hello there...

@SirIsaacTheRed 3 года назад

@@TomRocksMaths I like seeing Flammy here:) To be expected I guess.

@malignusvonbottershnike563 3 года назад

It's the end of the day on Christmas day, and I'm watching a maths video. It is honestly so incredible that I've finally reached a point where I can understand all the interesting stuff I've been seeing for months on the maths side of RU-vid, now that I've properly studied it. Thank you for videos like these, they're the reason I love maths so much.

@TomRocksMaths 3 года назад

This is amazing to read - well done and keep it up :)

@mehulchakraborty_0517 3 года назад

I find you very expressive, like my maths teacher never used to move the hands, speak in an enthusiastic way, rather he had a machine voice and was dead serious . I discovered you today and I really admire your style of teaching

@TomRocksMaths 3 года назад

Thanks and welcome to the channel :)

@meiwinspoi5080 3 года назад

refreshingly different. loved it. wondering why you did not utter the word “granmmmaaa” function.

@TomRocksMaths 3 года назад

Thanks!! I talk about the Gamma function in great detail here in fact: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-7y-XTrfNvCs.html

@patrickhehl9881 3 года назад

I think this is one of the first math result where my eyes legitimately widened when the final box was drawn around the n! equation (and I'm a final year engineering student so I've seen plenty of neat results!) Thanks for the crystal clear explanation Tom - I had seen this formula before but the derivation really made it click. All the best from McMaster University in Canada.

@TomRocksMaths 3 года назад

Thanks Patrick :)

@GrassZeplin 2 года назад

I am a first year engineering student at McMaster rn. I found it interesting as well

@theleastcreative 3 года назад

his disrespect of dx's hurts my feelings

@MarcinSzyniszewski Год назад

I was screaming 🤣

@kelumo7981 Год назад

He's sloppy

@DaveJ6515 Год назад

Exactly. I was eagerly waiting for him to realize that he just put an equal sign between a differential and a function.

@bozzigmupp510 Год назад

@@DaveJ6515 What you mean

@lih3391 Год назад

@@bozzigmupp510 7:41 integration by parts

@gavcooper 3 года назад

This is great. I feel like he talks about this very briefly in 'Surely You're Joking, Mr Feynman' but I've never seen an explanation of it that I could particularly get my head around until now. Thanks!

@TomRocksMaths 3 года назад

Glad you enjoyed it Gav!

@astroandriodrox2356 Год назад

I loved your the explanation to this example. I’m sure any electrical engineers or maths majors watching will immediately think of how similar this example is to the Laplace transform, which uses similar proofs for some of the standard transforms.

@theimmux3034 3 года назад

What a way to link teaching a new integration method with something interesting in math. I had heard of the Gamma function and its sibling before but had never quite understood how they were exactly connected to factorials.

@TomRocksMaths 3 года назад

Glad you enjoyed it :)

@rjmorpheus 3 года назад

My mind is blown....this deserves a follow!

@TomRocksMaths 3 года назад

@oom_boudewijns6920 Год назад

best video i've watched in august !

@woodrow-1319 3 года назад

the way he makes calculus interesting and not boring to a high school kid that hasn't done anything past the required algebra classes is amazing. I literally understand nothing that is happening but it must be really cool if you understand it!

@TomRocksMaths 3 года назад

@jamesjohnson2394 2 года назад

As long as you have the excitment about learning I guarentee you'll become a master in anything you wanna do!

@tomgargan8339 4 месяца назад

Wow, such a simple and amazing trick. Thanks for such a great explanation

@robertschlesinger1342 2 года назад

Interesting and worthwhile video.

@football4773 3 года назад

Beautifully done....clearly explained

@TomRocksMaths 3 года назад

Glad you liked it Suhas :)

@karimalramlawi7228 4 месяца назад

This is also called the laplace transform of x^n when s=1

@owen7185 Год назад

Absolutely brilliant video

@miraeklund7597 3 года назад

Thank you so much for your work :)

@TomRocksMaths 3 года назад

My pleasure Filip :)

@YorangeJuice 2 года назад

wow that was amazing thank you

@ricardozabalayoe2672 Год назад

Very well explained man!

@KlausDieckmann 2 года назад

Well explained.

@noonesperfect 3 года назад

Great explanation .... interesting story , Feynman itself is sure special kinda genius 👍

@bartgillis4352 2 года назад

very nice explanation. good job 👍

@numbers93 Год назад

This is beautiful

@daphenomenalz4100 3 года назад

My teacher of my tution during my school actually taught this and never told us it was feynmann's technique, he just said it's a better of doing integral that are similar to integrals you know already. He also said that it won't be asked in exams but still learning it would be helpful.

@neruneri 3 года назад

This is beyond my understanding and education level, but you're such a good presenter that I'm happily watching nonetheless. Have to say, I greatly admire you and your enthusiasm in communicating maths! While I struggle to follow along with maybe even most of your videos, some of them have nonetheless made it click for me and I've come away with more understanding than I had before, and that is deeply appreciated!

@TomRocksMaths 3 года назад

This is amazing to hear - thank you

@521Undertaker 3 года назад

Someone better put out an APB for all those missing differentials.

@vasuhardeo1418 3 года назад

the drawings made me smile , what fun

@TomRocksMaths 3 года назад

Ben's drawings are awesome

@ashishrathorear7x 8 месяцев назад

Respected Dr. Tom! You always remind me of my one friend who left me after the graduation. 🥺

@chivoronco4853 10 месяцев назад

good explanation

@johnchessant3012 3 года назад

One of my favorite "differentiation under the integral sign" problems is sin(x)/x, from 0 to infinity. You do it by defining I(a) = int sin(x) exp(-ax) / x. Then differentiate w.r.t. a, which gives a closed-form for I'(a), which you then integrate to find I(a). Finally set a = 0.

@TomRocksMaths 3 года назад

Love it!

@imnotdeadinsideiswear2002 3 года назад

this was soooooo satisfying im literally bursting with happiness

@TomRocksMaths 3 года назад

@Lucky10279 3 года назад

This so neat!

@TomRocksMaths 3 года назад

Glad you enjoyed it!

@nishatmunshi4672 3 года назад

I really enjoyed

@TomRocksMaths 3 года назад

Awesome

@adaircampos4240 3 года назад

Great video!

@TomRocksMaths 3 года назад

Thanks Adair!

@pardobrayan3130 3 года назад

From Colombia, I admire you :').

@TomRocksMaths 3 года назад

Amazing thanks Pardo :)

@Thror251 3 года назад

you can also solve this integral with induction by starting at n=0 and proving the induction step.

@TomRocksMaths 3 года назад

Nice spot.

@bachirblackers7299 3 года назад

Brilliant .

@TomRocksMaths 3 года назад

Thanks Bachir!!

@Taterzz 2 года назад

i am jealous of that buttery smooth handwriting.

@eng560 3 года назад

Nice explanation

@TomRocksMaths 3 года назад

Glad you enjoyed it Yusuf :)

@cheasify 3 года назад

Useful thanks

@TomRocksMaths 3 года назад

Happy to help :)

@johnchessant3012 3 года назад

Here's an even slicker way to integrate x^n exp(-x): Let a be a parameter with -1 < a < 1. First, integrate exp((a-1)x) dx from 0 to infinity, and expand the result as a power series in a (it's a geometric series). Then, expand exp((a-1)x) as a power series in a, and integrate the result term-by-term dx. Now compare coefficients!

@TomRocksMaths 3 года назад

These are great John - thanks for sharing :)

@sunandinighosh6037 2 года назад

This is somehow incredibly beautiful 😌

@davidbrisbane7206 3 года назад

I've seen quite a few uses of Feyman's integration technique. What is never discussed is the conditions under which this method works. I.e it is never stated and it is never verified.

@caelanpereira5458 3 года назад

Exactly It would be really useful to know straight away when this is applicable

@timotheehrb2481 3 года назад

Awesome 👍👍

@Joshua-hf5nl Год назад

Y'know, I came here for the bad drawings, and I was delivered the bad drawings. Thank you and well done!

@srinandanasastry3001 3 года назад

Superb.....👌👌🔥🔥

@TomRocksMaths 3 года назад

Thanks 🤗

@rossg9361 2 года назад

Feynman did invent this method. He popularized it.

@generalpartridge7653 3 года назад

This is well cool, loved your new video on the atomic bomb equation on numberphile too! Really nice trick now we know how to do it, integration is more knowing all these tricks and how to apply them eh? Cool stuff though.

@TomRocksMaths 3 года назад

Absolutely - integration is all about having as many tools as possible in your toolbox.

@reu.mathematicsacademy8566 Год назад

Feynman trick is the best

@coreymonsta7505 3 года назад

I guess only needed to do parts in its entirety for one step because, you found a formula right away

@rohansharma392 3 года назад

Sir I am from india and I like your work and love it

@pinklady7184 3 года назад

I am from Ireland and I love Indian dishes. 😋

@antonbordwine 3 года назад

I couldn't handle it so I subscribed just to see you more often. 😌

@TomRocksMaths 3 года назад

Welcome :)

@antonbordwine 3 года назад

@@TomRocksMaths :D

@whywouldyousub5472 7 месяцев назад

That drawing is in my school workbook

@akbarzamani9538 Год назад

good

@drvanhelsingz5133 Год назад

6:09, daammm I’m feeling a bit attacked right now 😂

@gresach 3 года назад

nice handwriting

@TomRocksMaths 3 года назад

@TommyTypewriter 3 года назад

Hey Tom, nice Viedo ! Could you maybe explain when it is formally okay to change the Integral and the differentiation?

@TomRocksMaths 3 года назад

It's called 'Leibniz Rule' and tells you that a 'full' derivative outside of the integral becomes a 'partial' derivative inside.

@darkdevil905 3 года назад

Thats a very cool trick but the trade off is that you need some good insight to alter the integrand such that you can get something which is the general case of your integrand that when differentiated generates your result.

@TomRocksMaths 3 года назад

It is indeed only applicable to some integrals, but its always nice to have more techniques to add to your integration toolbox :)

@caelanpereira5458 3 года назад

This is such a cool trick, but it seems to be tailor made to work only in quite specific cases, does anyone know if there are general integral types for which this technique always works? similar to an identity of sorts I guess

@DistortedV12 3 месяца назад

This was great. Your teaching style resonates with me more than 3Blue1Brown, ngl

@paulg444 Год назад

nice video, but actually the infinitesimal dv =e ^-x dx... these little things are important, they keep things clear.

@baptistebauer99 3 года назад

Sounds like Γ(x) with extra steps Brilliantly explained though. I wonder what kind of problem were those scientists having that took them 3 months to think about this trick, figured it out myself in a Laguerre Polynomials problem in about an hour and a half. I'm just learning now that this method is the Feynman Integral Trick

@sohammakim9178 Год назад

I tried integrating with IBP and skipping the times one in n!. Basically I tried integrating n!*(0-infinity)(x*e^-x). Using IBP you get that the answer is infinity*n!. I might have done something wrong but I don't think I did. I think this result comes about because the n! in the problem I gave wasn't actually n!, it was n! without the times 1 term.

@quidam3810 2 года назад

Great video, thanks a lot ! Does anyone know what was the integral that stopped the guy at Los Alamos ? Who were probably quite bright people !

@kyanvanuffelen1756 11 месяцев назад

only correct if n is a positive integer

@nugrars1253 3 года назад

I’m 37 and bad in math, but i’m still watching this 😂

@TomRocksMaths 3 года назад

As long as you're having fun :)

@maxwellsequation4887 3 года назад

37 is prime :|

@federicopagano6590 Год назад

this has an incredible symmetry at first glance too long to post here

@kasparwilliams2301 3 года назад

didn't need the ads mate great video tho

@alphalunamare 3 года назад

That was pretty cool :-)

@TomRocksMaths 3 года назад

I know right?

@alphalunamare 3 года назад

@@TomRocksMaths I just watched it again :-) It's like taking a short cut through the forest.

@vasuhardeo1418 3 года назад

that was cool

@TomRocksMaths 3 года назад

I know right?

@vasuhardeo1418 3 года назад

@@TomRocksMaths Yup, Phys and math is just to cool, but abstract math i still dont get group theory

@griffin7416 3 года назад

We can also use Gamma integration method 🙂

@TomRocksMaths 3 года назад

Yes, definitely

@Jkauppa Год назад

try this: elliptic integral of the 2nd kind, integrate sqrt(1+c*sin^2(z) ] dz = (2/3)*csc^2(z)*(c*sin^2(z)+1)^(3/2), the c-value is negative, csc has a zero-point issue

@Jkauppa Год назад

try standard integration in parts

@Jkauppa Год назад

same goes for the inverted ^-1 version integral

@rohansharma392 3 года назад

Sir can you explain integral by parts in geometry tipes can

@whatelseison8970 2 года назад

At 1:33 you forgot the closing parenthesis in the integral equation for 1/(1+x^3)! (That's an exclamation, not a factorial lol.) When I realized it I made that same face as the bad drawing.

@txikitofandango Год назад

The little drawn guy missed many chances to remind you to put in the missing "dx" as in "dv = e^(-x) dx", but nonetheless I loved this presentation

@gegebenein.gaussprozess7539 9 месяцев назад

This is a very important and correct comment. I agree 100%. If you rock maths, you should do it right, since maths won't enjoy it.😀

@mastershooter64 3 года назад

0:41 I like to think of derivatives and integrals like this taking the derivative of a function is like taking something apart, you can use anything and go to any extent, using a screwdriver, a hammer, or a bomb. But taking the integral of a function is like perfectly putting something back together, for the former you don't need much skill, for the latter you need to be much more skilled.

@TomRocksMaths 3 года назад

nice metaphor!

@adayah2933 Год назад

How do we know that it is correct to push differentiation under the integral sign? The hypotheses of Leibniz rule (finite bounds) are not met here.

@wiatraktymoteusz1328 2 года назад

It's very clearly explained but I am not sure how to chose an integral that would lead you to the integral that you want to calculate. is it always int(e^(-ax))?

@frederickwong4390 8 месяцев назад

I could only see repeated IBP to get the reduction formula - this is standard method found in literally every textbook. I don't see Feynman's method anywhere. Did I miss something?

@andrejgrebenc3235 Год назад

Hi presenter, give pls cases of indefinite integrals using Feynman method.

@Robo983 3 года назад

This is nitpicky, but shouldn't dv = e^(-x) * dx ? (also du should have a dx)

@zaheercoovadia4745 3 года назад

had the same issue haha

@lm58142 3 года назад

Thanks for sharing. At 2:09, that's a gamma function (of n+1), right?

@TomRocksMaths 3 года назад

Yes, exactly :)

@frozenmoon998 3 года назад

I smell the papa flammy, but also the papa tom :D - nice video on the Feynmann technique.

@TomRocksMaths 3 года назад

😋

@jamesbarrett1583 Год назад

Hi. I have a question. How was the non integer factorial calculated? Thanks, Jim.

@athul_c1375 3 года назад

on other note I understand Gamma function

@madvexing8903 3 года назад

I'm confused - why when e^-ax is differentiated, how can you differentiate with respect to a, as seen at 10:21, even though the integral says dx at the end? Apologies if this is a stupid question.

@TomRocksMaths 3 года назад

We are doing a partial derivative which means that we are saying the function e^(-ax) is in fact a function of two variables so f(x,a) = e^(-ax), and then we can differentiate with respect to a only. This doesn't affect the integral because it is with respect to the variable x as you say. I'd suggest taking a look ay my video on partial differentiation if this idea is new to you: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-RVwcBGzQcT8.html

@Arthur0000100 3 года назад

What is the integral he solved at Los Alamos??

@Zoxesyr 3 года назад

Tom, If you are going to insult Americans, at least take your shirt off.

@rajinfootonchuriquen 3 года назад

No homo

@edmundwoolliams1240 Год назад

I think it’s important to note, which I think Tom did in this video, that the ‘trick’ wasn’t invented by Feynman. He just famously used it in many important applications, having read about it in a calculus book.

@maindepth8830 3 года назад

This looks very interest8ng but i have no clue what any of this means

@TomRocksMaths 3 года назад

Well as long as you're having fun!

@maindepth8830 3 года назад

@@TomRocksMaths thanks

@jeromeheaven5556 2 года назад

I don’t like how loosely the improper is treated. It’s important to emphasize that an improper integral is a limiting process where convergence is key! However, that was clearly not the point of this video. Fun exposition, nonetheless. For those of you wanting to know, generally, when and why this works, look up Leibniz Integral Rule. In a nutshell, continuity of a function of two variables and a certain partial derivative explains when you can interchange the order of partial differentiation and integration.

@irstafoto Год назад

Missing right parenthesis at 1:33

@FemboyConquest 2 года назад

15 year old me: Ewww maths. 20 year old me: Fascinating, I understand nothing, but by god is this amazing!

@manuelocana8074 2 года назад

why a though?

@ajokaefi 3 года назад

at 2:56 you must use exterior differentiation dv= bla bla bla dx (dont forget dx)

@hectorbello7976 3 года назад

Nice explanation, just one doubt, Is it really Feynman's work or JulianSchwinger (parametrization) result? any publication you know that gives the price to the winner?

@TomRocksMaths 3 года назад

The story comes from Ben's book so I would suggest checking it out: www.amazon.co.uk/Change-Only-Constant-Wisdom-Calculus-ebook/dp/B07PRHKN1X