Feynman's Technique of Integration

Feynman's trick for integration, aka differentiation under the integral sign. This integration technique is very useful in calculus and physics.
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[@blackpenredpen]

Is it fish or alpha?

[@mrakoslav7057]

Use something like Ž than you cant mess up

[@ssdd9911]

alfish

[@chirayu_jain]

Maybe *alpha fish* 😅

[@pwnd785]

Fish of course

[@takureido3122]

It is *a* fish

[@tjdowning4263]

You could also write the cos term as the real part of e^i5x, and then complete the square in the exponential to get the final answer. Physicists use that trick a lot in quantum field theory.

[@michalbotor]

f(a) := integral from 0 to oo of exp(-x^2) cos(ax) dx g(a) := integral from 0 to oo of exp(-x^2) sin(ax) dx H(a) := integral from 0 to oo of exp(-x^2) exp(iax) dx H(a) = f(a) + ig(a) ∴ f(a) = Re(H(a)) && g(a) = Im(H(a)) ------------------------------------------------------------------------------------- exp(-x^2) * exp(iax) = exp( -x^2 + iax ) = exp(-( x^2 - iax )) = exp(-( x^2 - 2(ia/2)x + (ia/2)^2 - (ia/2)^2 )) = = exp(-( (x - ia/2)^2 + a^2/4 )) = exp( -(x - ia/2)^2 - a^2/4 ) = exp(-(x - ia/2)^2) exp(-a^2/4) ------------------------------------------------------------------------------------- H(a) = integral from 0 to oo of exp(-(x - ia/2)^2) exp(-a^2/4) dx = = exp(-a^2/4) integral from 0 to oo of exp(-(x - ia/2)^2) dx ------------------------------------------------------------------------------------- i am stuck at this moment. i tried the transformation u := x - ia/2 but i don't know what to do with the integral: integral from -ia/2 to (oo - ia/2) of exp(-u^2) du that has complex limits (i don't know if that is how i was supposed to set the limits of u variable either) and I am not able to split it into two integrals of real variable either. can you give me a hint how can i proceed from here?

[@still.sriracha]

@@michalbotor you did all that before understanding the basic concept of substitution :) Exp(-x^2) if multiplied by the euler's theorem would lead to addition of i in the expression whose integral in forward solving is a pain in butt (from past experiences) So moral is to find a logical concept and think on it before just scribbling this is pro tip in competitive level prep. Be well my friend.

[@tanmaymishra9576]

Trueee

[@groscolisdery1158]

I was going to point it out as my way. But, I guess, the hosts wants to teach the Feynman's method. By the way, Feynman was a physicist if I remember correctly.

[@groscolisdery1158]

try y =x+-alpha*x/2

[@krukowstudios3686]

Wow... an integral question solved by partial derivatives, integration by parts, differential equations and the Gaussian Integral to top it all off. Amazing! More Feymann technique questions, please!!

[@ekueh]

Wow! Feyman’s technique, DI method, Gaussian, ODE all in one. What else can top this? Adding a bit of FTC perhaps

[@cpotisch]

It inherently involves FTC because it involves indefinite integrals.

[@executorarktanis2323]

What's the full form of ftc?

[@BiscuitZombies]

AND the chen lu

[@cpotisch]

@@executorarktanis2323 Fundamental Theorem of Calculus. Which there already was plenty of, so I don’t see how OP thinks it was missing.

[@executorarktanis2323]

@@cpotisch oh thanks this brings back memories from when I was trying to learn calculus by youtube (self learnt) and didn't know the terms thanks for explaining it now since now I have more broad understanding than what I did 3 months ago

[@ashwinmurali1911]

This is the coolest thing I watched today

[@thatkindcoder7510]

The coolest thing so far

[@GusTheWolfgang]

That's insane!!!!!!!!!!!!!!!!!!!! I love it. It makes me sad they don't teach this in my engineering courses :(

[@blackpenredpen]

AugustoDRA : ))) I actually didn’t learn this when I was in school too. Thanks to my viewers who have suggested me this in the past. I haven a video on integral of sin(x)/x and that’s the first time I did Feynman’s technique.

[@SimsHacks]

It's covered in measure theory (math majors only) as one of the conditions to use the theorem is to find a L¹ function such that |d/da f(x,a)| ≤g(x) for almost all x. L¹ = set of functions with finite Lebesgue integral (not ±∞)

[@maalikserebryakov]

If you’re sad about that, you don’t belong in engineering. arcane mathematical techniques are nothing but a tool to an engineer, the primary of objective of an engineer is the creative process of ideating new machine designs, and this on its own is a massively difficult issue that takes enormous creative power. If you’re focusing on learning esoteric integration techniques, you aren’t focusing on engineering. I bet you aren’t an engineer now.

[@GusTheWolfgang]

@@maalikserebryakov hahaha, you hit the nail on the head.

[@thesnackbandit]

@@GusTheWolfgang Was he right?

[@MiroslawHorbal]

I really wish youtube existed when I was studying mathematics. The potential to be educated in advanced topics without paying a hefty fee for university tuition will hopefully change this world for the better.

[@chirayu_jain]

If nothing works to solve a integral Then *feynman technique* would work😉 BTW in the description of book, your name was also there 😁

[@blackpenredpen]

Chirayu Jain yup! I gave a review of the book : )))

[@roswelcodiep.bernardo7288]

Not that much... Sometimes we need to use complex analysis which includes residue theorem or Cauchy's Theorem

[@felipelopes3171]

You can also notice that the function is even and replace the integral with half the integral from -inf to inf. Then you break up the cosine into two complex exponentials, separate into two integrals. For each one you can complete the square in the exponent and reduce to the integral of exp(-x^2) by shifting the variable.

[@denissmith7671]

Niceee 🤤

[@AlanCanon2222]

I found the book in college that Feynman learned this trick from, it's Advanced Calculus By Frederick Shenstone Woods · 1926.

[@TheArtmatician]

Wow, that's nice

[@chirayu_jain]

I remember this method, because in the video contest I did the integral of (e^-(x^2))*cos(2x) from 0 to infinity. BTW whenever I see e^(-x^2), I always think about feynman technique.

[@blackpenredpen]

Chirayu Jain Oh yea you did. And you did a great job on that. : )

[@mariomario-ih6mn]

I changed my profile picture recently

[@jumpman3773]

@@mariomario-ih6mn Me too

[@100kbeforetheyeartwothousa7]

@@jumpman3773 Hi

[@marinmaths3826]

Very nice use of Feynman’s technique. I’m getting the book rn!

[@blackpenredpen]

Very nice!! Thanks.

[@srki22]

I really enjoy your enthusiasm while explaining things :) Thank you for the videos and please, never lose the energy, liveliness, and passion that you have now. Very nice!

[@kingbeauregard]

My initial intuition was to use Feynman to get rid of the exponential term, because if you can get rid of that, trig functions are easy. The thing I didn't think through was the limits of integration: a trig function has no limit at infinity. So quite counterintuitively, it was the cosine that was going to be the troublesome element in all this, while the exponential term was what made the thing solvable.

[@raunakroybarman1027]

I am agreeing that Feynman's technique is having a good strong hold in solving exponential integrals...but rather than complicating we could have solved it by manipulating "cos(5x)" as (e^5ix + e^-5ix)..it also saves the time...

[@mekkinoureddine9647]

Maths with you are wounderfull, thanks

[@frenchimp]

It's a bit crazy to call that the Feynmann technique. It goes back to Leibniz and it"s just deriving an integral depending on a parameter. Which by the way demands justification (either uniform convergence or dominated convergence). And in order to make this work you have to be extremely lucky and have a good intuition because you need 1) to find the right parametrization (here it's pretty obvious) ; 2) to be able to integrate the partial derivative for each value of the parameter (which is most of the time not possible) 3) to end up with a differential equation which you can solve (which is most of the time impossible), 4) to be able to compute a special value (here you need to know the value of the Gaussian integral, which is in itself tricky). So, I'd say it's a nice trick when it works but doesn"t qualify as a method...

[@JohnSmith-qp4bt]

It looks like the this problem was purposely designed to arrive at an aesthetically pleasing solution. (Given all the justifications/special circumstances/restrictions you mentioned)

[@loudfare8840]

@@Hmmmmmm487Feynman learnt this method in a random book during his undergrad and he famously showed off to basically everyone that he could solve otherwise very hard integrals.

[@artempalkin4070]

Makes me a bit mad when people call it Feynman's technique. The guy did a lot of good things, but this one has nothing to do with him. They're basically saying that only an American in the middle of 20th century could come up with such idea... What did people all over the world do before that, when calculus was already so advanced, and things like FT and others were well known...

[@mikeheyburn9716]

As a teacher, I loved you saying "negative fish" and will use that in future. Cheers, always good to watch your videos too.

[@TechnoCoderz369]

This is great! Thank you! Richard Feynman really was a genius!

[@TechnoCoderz369]

The channel name is blackpenredpen but you also use blue pen

[@nithinsirimanne2924]

🫤🤡

[@mamadetaslimtorabally7363]

Excellent explanation. So brilliantly explained. Thanks a million.

[@akshatahuja2523]

All the feynman's techniques are UNIQUE 👍👍

[@geometrydashmega238]

I've read some of the book's reviews and it looks awesome. I might pick one soon, the applications and integration techniques look interesting

[@ChollieD]

This is such an elegant proof. Really impressive.

[@IshaaqNewton]

Sometimes, a lot of integral practices makes me to say Instagram as Integram

[@blackpenredpen]

lol!

[@NXT_LVL_DVL]

Instagram is the culprit

[@octopuspartyofficial]

When you set alpha equal to sqrt(2 - 4ln(2)), you get sqrt(pi / e) for the answer. Pure beauty indeed.

[@hhht7672]

This is the hardest integral I’ve gotten right on my own! So proud of myself

[@shivanshbellamkonda6898]

Integrate (e^x)(x^x)(2+logx) wrt to x Please someone do this

[@ericwilliams1832]

x^x oh no

[@deletedaxiom6057]

Well I must say ty to you Mr. @blackpenredpen . Thanks to your videos I finished Differential Equations with a B. It was on of my last 2 math classes for my mathematics BS

[@blackpenredpen]

Nice! I am very glad to hear! : )

[@michaelstudley8062]

I'm halfway through algebra 1, and yet somehow I understand and enjoy most of these videos. You and other channels like you (e.g. Mathologer) make this stuff really accessible, and importantly, fun. (Not to say I don't enjoy my algebra 1 class!)

[@skyrider8890]

A great teacher is everything, right?

[@jzanimates2352]

Congrats on 400K subscribers!!!

[@blackpenredpen]

JZ Animates thank you!!

[@BluesyBor]

DAYUM, that's one of the most elegant solutions I've ever seen! Why none of my professors was teaching this when I was studying?

[@tomctutor]

I notice the *Feynman' technique* (aka. _Leibniz Integral Rule_ ) depends basically upon parameterizing the parts expansion here; its the _by-parts_ part that gives it the power in my opinion for what its worth!

[@giovannimariotte4993]

Amazing¡¡¡¡ you must record more videos about this topic¡¡¡¡¡¡¡

[@cruzazul2609]

I don't have an advanced level of English but that's one of a lot of thing that I love Maths, it's an universal language and your passion in every video is the thing because of I'm still here. Imagine! If I can understand you and I don't speak English fluently, you're MORE THAN AMAZING. Lots of love from Mexicoooo ꒰⑅ᵕ༚ᵕ꒱˖♡

[@yashvardhan6521]

A beautiful technique explained beautifully!!

[@ralstonrobertson6644]

This was a unique derivation technique. Thank you for sharing.

[@johngillespie8724]

I like it. I love your enthusiasm too.

[@chetnarayan9156]

You didn't got views but all you got is alots of love from the lover of mathematics

[@mokouf3]

Feymann's Technique + Differential Equation

[@blackpenredpen]

Mokou Fujiwara yes. And Chen lu!

[@cameronbetts3902]

Because the function is even, you can take the integral from -infinity to infinity and then that would double your answer so the final answer (given alpha = 2) would just be sqrt(pi)/e which i think is even cooler

[@thaitrieu791]

Just saw the Gaussian integral=sqrt(pi)/2 half an hour ago in lecture hall. I didn’t know where it came from while my prof was explaining Laplace Transform of t^(-1/2). And now here… What a small world of Mathematics !

[@ardavalilable]

I love this channel!

[@blackpenredpen]

Thanks!

[@j121212100]

If there is anything I do not want to forget from my school days is it calculus. Such a beautiful form of math.

[@deeznutz-bn9sl]

POV: you can't sleep now, there are monsters nearby 7:36

[@stephenkormanyos766]

Beautiful. Thank you so much.

[@irvngjuarez]

That was beautiful man just phenomenal

[@thanasisconstantinou7442]

Such an elegant and clever integration technique. Bravo to Feynman and to you, of course. Very cool indeed.

[@epicmarschmallow5049]

Bravo to Feynman? For appropriating an integration technique known to Leibniz around 300 years earlier?

[@thanasisconstantinou7442]

@@epicmarschmallow5049 ?

[@prevostluc4025]

I love all your videos, they are hearwarming. Thank you so much !

[@WilEngl]

Nice and clean trick ! Thank you.

[@abhinavshripad5356]

π and e in a same expression is always beautiful

[@andrewandrus3296]

convert cosine to sum of exponentials, complete square, Gaussian integral is root pi, can do it in your head in a few minutes, under a minute if you're confident, and almost instantly if you've seen a few of these. Cool to see feynman's technique at work though, great video!

[@muddle.]

lovely video, it's this that makes me love calculus

[@physjim]

the sinx over exp x^2 when x goes to infinity needs a bit more rigor when calculating, you can't just say it's a finite number on the nominator (max +1 or min -1) because the lim of the sin function when x goes to infinity doesn't exist. I believe one way to alleviate this, is by using the "sandwitch" theorem; wikipedia -> Squeeze_theorem

[@elliotlacinai3041]

Sure but I think it's safe to assume that if the viewer understands Feynman integration, they also know (or intuitively understand, at the very least) why the expression evaluates to zero at inf

[@ingGS]

This is one of the most beautiful videos I have seen. ¡Very complete and engaging explanation!

[@nishukapil4038]

Can you integrate from 0 to inf ln(1+ix)*(1+ix)^(-b)/(e^(2*pi*x)-1)dx?

[@chadidridi9306]

University teacher: ur exam has integrals The intégral during the exam:

[@yurfwendforju]

Haven't seen a video for long time wich made me so happy :)

[@peter_vincent_vampire_hunter]

Absolutely elegant

[@samvaidansalgotra7427]

Beautiful explanation😀

[@cletoazzani7763]

Wow, nice solution !!!

[@davidm.johnston8994]

I wish I understood. Someday, maybe. Man that's orders of magnitude beyond what I can comprehend at the moment.

[@mokouf3]

This is an amazing question for Calc 2.

[@AlbrechtJ]

Thanks for remind me of a feeling in Maths again. It is 20+ years ago since my college course. Actually, I love Maths, in the past It is my recreation. But living in real life, I have no time to solve the challenge Maths problem. Life is hard.

[@PunmasterSTP]

I very much enjoy watching the derivations, even though I know I'd probably never be able to figure it out myself.

[@LorenzoWTartari]

A fun trick would also be using the fourier tramsform of the bell curve

[@matthieumoussiegt]

very good proof amazing use of differential equations

[@jacobbills5002]

Just watch this impressive Math channel kzbin.info/door/ZDkxpcvd-T1uR65Feuj5Yg

[@dudewaldo4]

That was an experience. What a crazy and amazing technique

[@xenolalia]

One can also observe that f(\alpha) is (up to a constant factor) just the Fourier transform of e^{-x^2}.

[@samuelglover7685]

Very nicely done! Thanks!

[@EntaroCeraphenine]

One of the best crossover episodes ever

[@thedoublehelix5661]

10:54 I was thinking he would let it equal i

[@pkvlogs5078]

hlo there ... i hd seen yewr question last night nd started solving right after .. well i wanna tell yew that i have expand COS( AX) to taylor series form :- summation(n€ (0 to infinity)) (-1)^n (A)^2n X ^(2n) / (2n)! then substitute the above into the integral and then bring out the summation [along with the constant term (-1) ^n A^(2n) / (2n) ! ] nd move the integral inside such that we will get integral (0 to infinity) x^(2n) e^((-x* x) i.e x square) dx now using gamma function definetion this integrand would be written like this 1/2 times gamma function (n+ 1/2 ) times gamma function (1/2) which is (pi)^(1/2) now (n+ 1/2 ) ! which can further be written like (n+1/2) ! = [(2n)! / 4^(n) (n) ! ] now substituting this result back into the summation we will see that the result we get is in the form 0.5 times gamma function(0.5) times summation (n €( 0 to infinity)) C^(n) / (n) ! where C = -A^(2) / 4 finally e^C = summation (n€ (0 to infinity) ) C^(n) / n! so the net result for this kinda integral will be {0.5 times gamma function(0.5) times e^ (- A*A /4 ) }

[@Mr_Mundee]

you can also use the taylor series for cos(5x) and use the gamma function

[@sumiransubedi6710]

11:38 I love how satisfied he looked after all that he did.

[@kushagragupta3416]

👀great work sir

[@shayakbardhan2064]

11:20 ...............the sentence is veryyy TRUE indeed !!!!

[@haradhandatta7048]

Very Nice.Thanks.

[@phill3986]

Nice don't remember running across the Feynman technique before.

[@chielvooijs2689]

Even more beautiful: Since e^(-x^2) cos(2x) is an even function, the integral from -inf to inf just becomes sqrt(pi)/e.

[@balajilakshminarayanan170]

such a beautiful video thanks

[@agrajyadav2951]

this made my day

[@hanst7218]

Great video man!

[@sameernd]

I HAVE A QUESTION > when the a=5 the C2 did not = to the C2 when the a=0 ! i think so

[@therealbazor]

Awesome vid, really enjoyed it!!!!!!

[@gevodem]

These integrals show up quite often in quantum mechanics.

[@ProCoderIO]

Apparently that’s why it’s “Feynman’s”. Universities didn’t teach it. He learned it from an obscure textbook. Hopefully more are teaching it today.

[@emiliomontes2043]

Awesome, can u show us the demonstration? thanks !

[@EMEngi4ALL]

Please tell me what calculus book are you using

[@scienceapps1467]

Can be done directly using chain rule of differentiation

[@integralbilmeyenfizikmezun111]

Thank you guy.

[@oyoyoyo]

This is a direct application of the Leibniz integral rule. Feynman may have rediscovered it by himself, but it is more a trick since it is a theorem.

[@aserioussalamander9475]

I don't understand it, but I can tell from his excitement that this is some pretty profound shit right here.

[@michaeledwardharris]

That was a wild ride!

[@antoniokokic7488]

Can't say I understand, but I do agree: it's very nice!

[@iR3vil4te]

I follow everything except how you justify that first step of bring the derivate inside the integral - what’s the basis behind that?

[@allapiagia]

“This is very very nice ! “

[@GP-sy9ei]

I have a number, x. I toss a coin. If it lands on heads, I double x. If it lands on tails, I add 1. What is the probability that, after n coin tosses, x is a multiple of 3?

[@kimothefungenuis]

you can also use a Fourier transform

[@Gungus-v1g]

Hi, very cool video! Although I did not understand one step. When we let alpha=0 to get rid of the cos(alphax), I don't understand how the constant C2 we found would also work when alpha=5. Any clarification would be greatly appreciated!

[@earthpcCHClS]

C2 is a constant that is independent of alpha. No matter what alpha would be, c2 would be the same. It's a similar process to solving for constants with ordinary differential equations