Feynman's Technique of Integration

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  • čas přidán 14. 12. 2019
  • Feynman's trick for integration, aka differentiation under the integral sign. This integration technique is very useful in calculus and physics.
    Subscribe to ‪@blackpenredpen‬ for more fun calculus videos!
    Check out the book, Advanced Calculus Explored, amzn.to/2PpOJIX
    Check out daily_math_, / daily_math_

Komentáře • 602

  • @blackpenredpen
    @blackpenredpen  Před 4 lety +1600

    Is it fish or alpha?

  • @tjdowning4263
    @tjdowning4263 Před 4 lety +698

    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
      @michalbotor Před 3 lety +1

      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
      @still.sriracha Před 3 lety +26

      @@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
      @tanmaymishra9576 Před 2 lety +1

      Trueee

    • @groscolisdery1158
      @groscolisdery1158 Před 2 lety +8

      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
      @groscolisdery1158 Před 2 lety

      try y =x+-alpha*x/2

  • @krukowstudios3686
    @krukowstudios3686 Před 4 lety +429

    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
    @ekueh Před 4 lety +559

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

    • @cpotisch
      @cpotisch Před 3 lety +39

      It inherently involves FTC because it involves indefinite integrals.

    • @executorarktanis2323
      @executorarktanis2323 Před 3 lety +2

      What's the full form of ftc?

    • @BiscuitZombies
      @BiscuitZombies Před 3 lety +2

      AND the chen lu

    • @cpotisch
      @cpotisch Před 3 lety +17

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

    • @executorarktanis2323
      @executorarktanis2323 Před 3 lety

      @@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
    @ashwinmurali1911 Před 4 lety +160

    This is the coolest thing I watched today

  • @GusTheWolfgang
    @GusTheWolfgang Před 4 lety +290

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

    • @blackpenredpen
      @blackpenredpen  Před 4 lety +79

      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
      @SimsHacks Před rokem +5

      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
      @maalikserebryakov Před rokem +8

      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
      @GusTheWolfgang Před rokem +4

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

    • @thesnackbandit
      @thesnackbandit Před 9 měsíci +2

      @@GusTheWolfgang Was he right?

  • @srki22
    @srki22 Před 4 lety +8

    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!

  • @chirayu_jain
    @chirayu_jain Před 4 lety +149

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

    • @blackpenredpen
      @blackpenredpen  Před 4 lety +18

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

    • @roswelcodiep.bernardo7288
      @roswelcodiep.bernardo7288 Před 2 lety +1

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

  • @MiroslawHorbal
    @MiroslawHorbal Před 4 lety +13

    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.

  • @AlanCanon2222
    @AlanCanon2222 Před 2 lety +6

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

  • @felipelopes3171
    @felipelopes3171 Před 4 lety +44

    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.

  • @prevostluc4025
    @prevostluc4025 Před 4 lety

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

  • @chirayu_jain
    @chirayu_jain Před 4 lety +61

    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.

  • @yashvardhan6521
    @yashvardhan6521 Před 3 lety

    A beautiful technique explained beautifully!!

  • @geometrydashmega238
    @geometrydashmega238 Před 4 lety +4

    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
    @marinmaths3826 Před 4 lety +9

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

  • @mekkinoureddine9647
    @mekkinoureddine9647 Před 4 lety +3

    Maths with you are wounderfull, thanks

  • @mamadetaslimtorabally7363
    @mamadetaslimtorabally7363 Před 6 měsíci

    Excellent explanation. So brilliantly explained. Thanks a million.

  • @ralstonrobertson6644
    @ralstonrobertson6644 Před 2 lety

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

  • @frenchimp
    @frenchimp Před 2 lety +32

    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
      @JohnSmith-qp4bt Před 2 lety +2

      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
      @loudfare8840 Před 6 měsíci +1

      @@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
      @artempalkin4070 Před 13 dny

      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...

  • @samuelglover7685
    @samuelglover7685 Před 4 lety

    Very nicely done! Thanks!

  • @kingbeauregard
    @kingbeauregard Před 4 lety +9

    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.

  • @TechnoCoderz369
    @TechnoCoderz369 Před 10 měsíci +1

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

  • @ingGS
    @ingGS Před 4 lety

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

  • @johngillespie8724
    @johngillespie8724 Před 2 lety

    I like it. I love your enthusiasm too.

  • @IshaaqNewton
    @IshaaqNewton Před 4 lety +32

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

  • @ChollieD
    @ChollieD Před 4 lety

    This is such an elegant proof. Really impressive.

  • @deeznutz-bn9sl
    @deeznutz-bn9sl Před rokem +5

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

  • @stephenkormanyos766
    @stephenkormanyos766 Před 4 lety

    Beautiful. Thank you so much.

  • @WilEngl
    @WilEngl Před 4 lety

    Nice and clean trick ! Thank you.

  • @dudewaldo4
    @dudewaldo4 Před 4 lety

    That was an experience. What a crazy and amazing technique

  • @mokouf3
    @mokouf3 Před 4 lety +10

    Feymann's Technique + Differential Equation

  • @giovannimariotte4993
    @giovannimariotte4993 Před 4 lety

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

  • @mokouf3
    @mokouf3 Před 4 lety

    This is an amazing question for Calc 2.

  • @mikeheyburn9716
    @mikeheyburn9716 Před 10 měsíci +2

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

  • @BluesyBor
    @BluesyBor Před 4 lety

    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?

  • @therealbazor
    @therealbazor Před 4 lety

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

  • @balajilakshminarayanan170

    such a beautiful video thanks

  • @muddle.
    @muddle. Před 8 měsíci

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

  • @user-pm9il1mu4c
    @user-pm9il1mu4c Před 2 lety

    Beautiful!

  • @cameronbetts3902
    @cameronbetts3902 Před 4 lety +1

    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

  • @andrewdouglas793
    @andrewdouglas793 Před 3 lety

    Absolutely elegant

  • @zubmit700
    @zubmit700 Před 4 lety

    This was really nice!

  • @samvaidansalgotra7427
    @samvaidansalgotra7427 Před 3 lety

    Beautiful explanation😀

  • @integralbilmeyenfizikmezun111

    Thank you guy.

  • @pjhh8798
    @pjhh8798 Před 9 měsíci

    beautiful, thank you

  • @sotocsick3195
    @sotocsick3195 Před 20 dny

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

  • @jzanimates2352
    @jzanimates2352 Před 4 lety +1

    Congrats on 400K subscribers!!!

  • @phecdu
    @phecdu Před 4 lety

    Love this. Like magic. So beautyful 😱

  • @hanst7218
    @hanst7218 Před 4 lety

    Great video man!

  • @irvngjuarez
    @irvngjuarez Před rokem

    That was beautiful man just phenomenal

  • @ardavalilable
    @ardavalilable Před rokem +1

    I love this channel!

  • @deletedaxiom6057
    @deletedaxiom6057 Před 4 lety +1

    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

  • @michaelstudley8062
    @michaelstudley8062 Před 4 lety +10

    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
      @skyrider8890 Před 4 lety +3

      A great teacher is everything, right?

  • @haradhandatta7048
    @haradhandatta7048 Před 4 lety

    Very Nice.Thanks.

  • @johnhumberstone9674
    @johnhumberstone9674 Před 4 lety

    Just beautiful!

  • @agrajyadav2951
    @agrajyadav2951 Před rokem

    this made my day

  • @cruzazul2609
    @cruzazul2609 Před 4 lety +3

    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
    @cletoazzani7763 Před 3 lety

    Wow, nice solution !!!

  • @raunakroybarman1027
    @raunakroybarman1027 Před 4 lety +33

    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...

  • @akshatahuja2523
    @akshatahuja2523 Před 4 lety +8

    All the feynman's techniques are UNIQUE 👍👍

  • @abhinavshripad5356
    @abhinavshripad5356 Před 4 lety +1

    π and e in a same expression is always beautiful

  • @nikunjy
    @nikunjy Před 2 lety

    Very nice !

  • @LorenzoWTartari
    @LorenzoWTartari Před 2 lety

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

  • @xenolalia
    @xenolalia Před 4 lety

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

  • @PunmasterSTP
    @PunmasterSTP Před rokem

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

  • @octopuspartyofficial
    @octopuspartyofficial Před 4 lety +4

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

  • @chadidridi9306
    @chadidridi9306 Před 4 lety +5

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

  • @EntaroCeraphenine
    @EntaroCeraphenine Před 4 lety

    One of the best crossover episodes ever

  • @hhht7672
    @hhht7672 Před 4 lety +39

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

  • @tomctutor
    @tomctutor Před 4 lety +7

    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!

  • @Patapom3
    @Patapom3 Před 4 lety

    Amazing!

  • @zaydabbas1609
    @zaydabbas1609 Před 3 lety

    This is such a pog method and this vid is amazing

  • @Mr_Mundee
    @Mr_Mundee Před 3 měsíci

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

  • @NO-vl8nq
    @NO-vl8nq Před 3 lety

    Thank u 💞

  • @thaitrieu791
    @thaitrieu791 Před 2 lety +1

    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 !

  • @chetnarayan9156
    @chetnarayan9156 Před 2 lety

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

  • @phill3986
    @phill3986 Před 4 lety

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

  • @thanasisconstantinou7442

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

  • @luanagodinho2340
    @luanagodinho2340 Před 4 lety +1

    Curto muito seu canal, você é fera! Brasil.

  • @rainbowiitneet6441
    @rainbowiitneet6441 Před 4 lety +2

    Great "sir"

  • @gourabpal5774
    @gourabpal5774 Před 2 lety

    Just wonderful 🤩

  • @davidm.johnston8994
    @davidm.johnston8994 Před 4 lety +1

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

  • @bazwardo7191
    @bazwardo7191 Před 3 lety

    This is amazing

  • @kushagragupta3416
    @kushagragupta3416 Před 3 lety

    👀great work sir

  • @michaeledwardharris
    @michaeledwardharris Před 2 lety

    That was a wild ride!

  • @matthieumoussiegt
    @matthieumoussiegt Před 3 lety

    very good proof amazing use of differential equations

    • @jacobbills5002
      @jacobbills5002 Před 3 lety

      Just watch this impressive Math channel czcams.com/channels/ZDkxpcvd-T1uR65Feuj5Yg.html

  • @kimothefungenuis
    @kimothefungenuis Před 7 měsíci

    you can also use a Fourier transform

  • @ernstuzhansky
    @ernstuzhansky Před 7 měsíci

    Wonderful

  • @j121212100
    @j121212100 Před rokem

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

  • @antoniokokic7488
    @antoniokokic7488 Před 2 lety

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

  • @emiliomontes2043
    @emiliomontes2043 Před 4 lety

    Awesome, can u show us the demonstration? thanks !

  • @markproulx1472
    @markproulx1472 Před 4 lety

    Fabulous!

  • @sergiolucas38
    @sergiolucas38 Před 2 lety

    Great video :)

  • @DynestiGTI
    @DynestiGTI Před 3 lety

    Very satisfying

  • @nishukapil4038
    @nishukapil4038 Před 4 lety +3

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

  • @seddikio1282
    @seddikio1282 Před 4 lety

    Very nice

  • @bruno68berretta53
    @bruno68berretta53 Před 3 lety

    Complimenti!

  • @shayakbardhan2064
    @shayakbardhan2064 Před 3 lety +1

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