Feynman's Technique of Integration

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4 жыл бұрын

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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Пікірлер: 604

@blackpenredpen 4 жыл бұрын

Is it fish or alpha?

@mrakoslav7057 4 жыл бұрын

Use something like Ž than you cant mess up

@ssdd9911 4 жыл бұрын

alfish

@chirayu_jain 4 жыл бұрын

Maybe *alpha fish* 😅

@pwnd785 4 жыл бұрын

Fish of course

@takureido3122 4 жыл бұрын

It is *a* fish

@tjdowning4263 4 жыл бұрын

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 3 жыл бұрын

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 3 жыл бұрын

@@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 2 жыл бұрын

Trueee

@groscolisdery1158 2 жыл бұрын

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 2 жыл бұрын

try y =x+-alpha*x/2

@krukowstudios3686 4 жыл бұрын

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 4 жыл бұрын

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

@cpotisch 3 жыл бұрын

It inherently involves FTC because it involves indefinite integrals.

@executorarktanis2323 3 жыл бұрын

What's the full form of ftc?

@BiscuitZombies 3 жыл бұрын

AND the chen lu

@cpotisch 3 жыл бұрын

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

@executorarktanis2323 3 жыл бұрын

@@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

@srki22 4 жыл бұрын

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!

@ashwinmurali1911 4 жыл бұрын

This is the coolest thing I watched today

@thatkindcoder7510 2 жыл бұрын

The coolest thing so far

@deeznutz-bn9sl Жыл бұрын

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

@GusTheWolfgang 4 жыл бұрын

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

@blackpenredpen 4 жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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

@thesnackbandit 10 ай бұрын

@@GusTheWolfgang Was he right?

@MiroslawHorbal 4 жыл бұрын

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.

@prevostluc4025 4 жыл бұрын

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

@chirayu_jain 4 жыл бұрын

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

@blackpenredpen 4 жыл бұрын

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

@roswelcodiep.bernardo7288 2 жыл бұрын

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

@geometrydashmega238 4 жыл бұрын

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

@marinmaths3826 4 жыл бұрын

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

@blackpenredpen 4 жыл бұрын

Very nice!! Thanks.

@AlanCanon2222 2 жыл бұрын

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

@ciahartsrodrigues8823 25 күн бұрын

Wow, that's nice

@stephenkormanyos766 4 жыл бұрын

Beautiful. Thank you so much.

@felipelopes3171 4 жыл бұрын

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 Жыл бұрын

Niceee 🤤

@yashvardhan6521 4 жыл бұрын

A beautiful technique explained beautifully!!

@mekkinoureddine9647 4 жыл бұрын

Maths with you are wounderfull, thanks

@samuelglover7685 4 жыл бұрын

Very nicely done! Thanks!

@ingGS 4 жыл бұрын

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

@user-pm9il1mu4c 2 жыл бұрын

Beautiful!

@TechnoCoderz369 Жыл бұрын

The channel name is blackpenredpen but you also use blue pen

@TechnoCoderz369 11 ай бұрын

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

@ralstonrobertson6644 2 жыл бұрын

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

@mamadetaslimtorabally7363 7 ай бұрын

Excellent explanation. So brilliantly explained. Thanks a million.

@kingbeauregard 4 жыл бұрын

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.

@johngillespie8724 2 жыл бұрын

I like it. I love your enthusiasm too.

@integralbilmeyenfizikmezun111 3 жыл бұрын

Thank you guy.

@ChollieD 4 жыл бұрын

This is such an elegant proof. Really impressive.

@WilEngl 4 жыл бұрын

Nice and clean trick ! Thank you.

@sotocsick3195 Ай бұрын

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

@frenchimp 2 жыл бұрын

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 2 жыл бұрын

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 7 ай бұрын

@@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 Ай бұрын

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...

@chirayu_jain 4 жыл бұрын

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 4 жыл бұрын

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

@mariomario-ih6mn 4 жыл бұрын

I changed my profile picture recently

@jumpman3773 4 жыл бұрын

@@mariomario-ih6mn Me too

@100kbeforetheyeartwothousa7 4 жыл бұрын

@@jumpman3773 Hi

@mikeheyburn9716 11 ай бұрын

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

@jzanimates2352 4 жыл бұрын

Congrats on 400K subscribers!!!

@blackpenredpen 4 жыл бұрын

JZ Animates thank you!!

@therealbazor 4 жыл бұрын

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

@agrajyadav2951 2 жыл бұрын

this made my day

@BluesyBor 4 жыл бұрын

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?

@IshaaqNewton 4 жыл бұрын

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

@blackpenredpen 4 жыл бұрын

lol!

@NXT_LVL_DVL 3 ай бұрын

Instagram is the culprit

@deletedaxiom6057 4 жыл бұрын

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 4 жыл бұрын

Nice! I am very glad to hear! : )

@chetnarayan9156 2 жыл бұрын

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

@hanst7218 4 жыл бұрын

Great video man!

@zubmit700 4 жыл бұрын

This was really nice!

@dudewaldo4 4 жыл бұрын

That was an experience. What a crazy and amazing technique

@nikunjy 2 жыл бұрын

Very nice !

@muddle. 9 ай бұрын

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

@balajilakshminarayanan170 4 жыл бұрын

such a beautiful video thanks

@pjhh8798 10 ай бұрын

beautiful, thank you

@andrewdouglas793 3 жыл бұрын

Absolutely elegant

@irvngjuarez Жыл бұрын

That was beautiful man just phenomenal

@johnhumberstone9674 4 жыл бұрын

Just beautiful!

@kushagragupta3416 3 жыл бұрын

👀great work sir

@ardavalilable Жыл бұрын

I love this channel!

@blackpenredpen Жыл бұрын

Thanks!

@NO-vl8nq 3 жыл бұрын

Thank u 💞

@mokouf3 4 жыл бұрын

This is an amazing question for Calc 2.

@Patapom3 4 жыл бұрын

Amazing!

@sergiolucas38 2 жыл бұрын

Great video :)

@samvaidansalgotra7427 3 жыл бұрын

Beautiful explanation😀

@giovannimariotte4993 4 жыл бұрын

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

@michaeledwardharris 2 жыл бұрын

That was a wild ride!

@cruzazul2609 4 жыл бұрын

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꒱˖♡

@cletoazzani7763 3 жыл бұрын

Wow, nice solution !!!

@thanasisconstantinou7442 4 жыл бұрын

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

@epicmarschmallow5049 2 жыл бұрын

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

@thanasisconstantinou7442 2 жыл бұрын

@@epicmarschmallow5049 ?

@michaelstudley8062 4 жыл бұрын

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 4 жыл бұрын

A great teacher is everything, right?

@phecdu 4 жыл бұрын

Love this. Like magic. So beautyful 😱

@DynestiGTI 3 жыл бұрын

Very satisfying

@haradhandatta7048 4 жыл бұрын

Very Nice.Thanks.

@chadidridi9306 4 жыл бұрын

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

@markproulx1472 4 жыл бұрын

Fabulous!

@sabysengupta3218 Жыл бұрын

fantastic

@zaydabbas1609 3 жыл бұрын

This is such a pog method and this vid is amazing

@ernstuzhansky 8 ай бұрын

Wonderful

@bruno68berretta53 3 жыл бұрын

Complimenti!

@cameronbetts3902 4 жыл бұрын

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

@bazwardo7191 3 жыл бұрын

This is amazing

@gourabpal5774 2 жыл бұрын

Just wonderful 🤩

@LeopoldBlooom 4 жыл бұрын

Nice!

@EntaroCeraphenine 4 жыл бұрын

One of the best crossover episodes ever

@yyc3491 3 жыл бұрын

Very very nice👍👍

@Mr_Mundee 4 ай бұрын

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

@vinayrana610 3 жыл бұрын

Oh man insane !

@alpharomeo3141 3 жыл бұрын

Bravo!!!

@octopuspartyofficial 4 жыл бұрын

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

@seddikio1282 4 жыл бұрын

Very nice

@sagulucky7245 Жыл бұрын

Very nice 👍👍

@matthieumoussiegt 3 жыл бұрын

very good proof amazing use of differential equations

@jacobbills5002 3 жыл бұрын

Just watch this impressive Math channel kzfaq.info/love/ZDkxpcvd-T1uR65Feuj5Yg

@rainbowiitneet6441 4 жыл бұрын

Great "sir"

@j121212100 Жыл бұрын

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

@PunmasterSTP Жыл бұрын

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

@xenolalia 4 жыл бұрын

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

@thaitrieu791 2 жыл бұрын

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 !

@abhinavshripad5356 4 жыл бұрын

π and e in a same expression is always beautiful

@user-wh7zx7wl4i 7 ай бұрын

Just.... Wow!

@raunakroybarman1027 4 жыл бұрын

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...