A RIDICULOUSLY AWESOME INTEGRAL: The best thing you are going to see this weekend!

[@blackpenredpen]

Lit af

[@chaos4785]

blackpenredpen Can u plz make a video of a different approach to this i integral😀i would like to see what you'll come up with❤

[@stevenxu5747]

Could this integral be solved more simply by contour integration? I was thinking that you can break the (sinx)^2 into e^(2ix) + e^(-2ix) - 2 /(-4). Close in the upper-halfplane for the first piece, bottom half-plane for the second piece, and either direction for the third piece. Observe that the integral along the real number line from -infinty to 0 is the same as from 0 to +infinity (as the function is even)

[@stevenxu5747]

Please make a video demonstrating contour integration, @blackpenredpen

[@vkilgore11]

Can you have the limits of integration be dependent on t as a consequence of using this technique?

[@JorgetePanete]

Beyonce Queen you forgot the question mark ;)

[@Wild4lon]

When you haven't learned Laplace transform yet but the symbols look cool

[@Jacob-ye7gu]

can confirm, laplace transforms are one of the coolest things you will learn

[@PapaFlammy69]

:)

[@trisanjitcreation143]

Please discuss details clear picture of your work. Don't hurry

[@homelesslukeskywalker7277]

@@Jacob-ye7gu do u live in Pittsburgh?

[@sagarmajumder7806]

Same pinch😇😇😇

[@trevr9924]

Dank. It was so nerve racking watching you make those two sign mistakes. When you caught them it was like watching a bomb being defused.

[@JorgetePanete]

who would win? -3 green chalkboards full of maths *-ONE MINUS SIGN BOI*

[@munendersingh5631]

Xd

[@yana_2_6_0]

Well negative 3 chalkboards would be pretty useless for solving

[@milanstevic8424]

@@yana_2_6_0 sounds like a challenge

[@xzy7196]

@@milanstevic8424 i cant click read more

[@chouhanaryan]

Bruh I ain't even finished high school and I have no idea why this popped up on my recommendations, but I watched the whole thing

[@munendersingh5631]

F

[@darkseid856]

Same

[@greensleeves6005]

He lost me at the Laplace stuff but he was too excited and I was too invested to stop watching

[@executorarktanis2323]

Me to

[@megacahh870]

Same. I enjoyed it tho.

[@samuelromero1763]

Using Laplace transforms for that is like bringing a shotgun to a game of paintball lol 😂

[@ernestomamedaliev4253]

@@MAN_FROM_BEYOND I have always been told that, hehe

[@CarlosSilva-ti5ib]

I agree. I solved it much more quickly using a standard method. By the way, the solution given on the vídeo is just one of the many solutions possible, wich appear naturally in the standard method.

[@chrisjfox8715]

Please make a video of the paintball solution 🙂

[@samuelromero1763]

@@chrisjfox8715 it’s just a second order differential equation, quite easy to solve using standard method. Just look up on khan academy.

[@CarlosSilva-ti5ib]

@@chrisjfox8715 I made one already, but in Portuguese.

[@ejvalpey]

I love how you can barely contain your enthusiasm.

[@gustavoespinoza7940]

“We’re going to use Leibniz rule” *physics majors want to know your location*

[@testusernameyoutube1]

gustavo espinoza i guess they call it feynman method 😬?

[@nombre3053]

laventiny feynman TRICK

[@rickhackro]

@@nombre3053 you meant METHOD

[@fym4x7]

No one: Absolutely no one: Not even Ramanujan: Papa Flammy: ok now we're gonna use the Laplace transform

[@davidblauyoutube]

"Not even Ramanujan" lmao

[@vandel_]

lol

[@marksaving756]

This took me about 2 minutes. Complex analysis gives an extremely elegant solution.

[@restitutororbis964]

Mark Saving Papa flammy likes to do it the cool way, without complex analysis. I bet its wayyyyyy easier with complex analysis (which I cant do yet so im forced to do it the cool way). Do you have any recommendations for any online sources or books I could get to learn complex analysis? I havent been able to find a good one.

[@categorille8330]

@@restitutororbis964 have you been able to find any? I'd be interested

[@restitutororbis964]

@@categorille8330 Yes I got a book on complex analysis a month ago. Its called complex variables and applications by Ruel V. Churchill. I havent gotten that far into it but its pretty good for an introductory book in complex analysis from what I've seen so far.

[@arbitrarilyclose]

Consult Papa Rudin.

[@AtotheKres]

Why use complex analysis if you can keep it real? See it as a training and the method he used was extremely cool and creative!

[@VisionaryModule]

I honestly went super saiyan when the differential equation showed up. best integral ever

[@nicholassignorelli132]

"The good thing is when doing partial fraction decomposition we just need to have the ability to read." - PapaFlammy You are my God.

[@JordanMetroidManiac]

You used differential equations as an integration method... that's incredible. Laplace transforms at that! You beat Wolfram Alpha lol

[@restitutororbis964]

Jordan Fischer Indeed. Laplace transforms are god like if you know when to use them and how to use them.

[@ericthegreat7805]

@alysdexia edgy

[@avananana]

This is what I mean when I tell my friends that mathematics is beautiful. Results like these, those that just pop out of the blue and completely makes you unable to talk, those are the best moments you'll ever have. Beautifully done and indeed, a very beautiful integral which definitely beautified my Saturday evening =)

[@broadcast3ful]

If those are the best moments you will ever have then you have one miserable ass life.

[@centralprocessingunit4988]

@@broadcast3ful your comment got two likes over three years. ==> successful fail.

[@TheGrimravager]

the enthousiasm, the clickbait title, this is the best thing ever

[@nejlaakyuz4025]

If its real it isnt clickbait

[@imacds]

ok so I wanted to refresh myself on both Laplace transforms and Feynman's method and was about to look both up separately but I thought to myself I will watch one more meme integral video... yet here i find myself. thank you

[@npip99]

There needs to be more people like you; salivating over Putnam problems alone is a sad life. I wish for a world where all KZbin thumbnails are as glorious as the one for this video

[@cuie6967]

21:53 : "am i that bad at math?, why is there +pi/2 instead of -pi/2" After two more hours and a handmade demonstration: "Let's see what he obtains as a final result" 23:49: "oh... I see....." :)))) Amazing video bro! Keep up the good work! This really made my weekend!

[@oni8337]

What's also ridiculously awesome is that the graph of sin²(x)/(x²(x² + 1)) has a very close fit to the normal distribution curve or e^-(x^2)

[@andrewhayes7124]

I'm in highschool AP calc and this just shows how much more I can still learn

[@calebanderson5309]

I'm putting off homework I need to do for homework I want to do. Your enthusiasm is what really makes this video great

[@AndriiMalenko]

21:58 its amazing that you solved differential equation but missed the "minus" sign near the first term: J(t) = -\frac \pi 2 e^{-2t} + \frac \pi 2

[@vadimkhudiakov526]

Second derivative I''(t) can be calculated through residues at (-i, +i) in Complex plane. Thus, I(t) = C1 + C2*t +pi/4 exp(-2t). If we estimate (sin(tx)/x)^2

[@ShaneClough]

You have the most infectious enthusiasm I've ever seen displayed. Subbed!

[@titarch]

I can't understand why you're not a +1M subscribers channel, your content is so great, so satisfying to watch, both funny and educative. People just don't get what's truly beautiful I suppose.

[@OGUZHANKOSARMD]

Loved your German accent. sounds like some Swiss.

[@1anya7d]

Sin(y) = y^(0r+m*0*m)/g(a*y)

[@cameron6751]

LanYarD N'(o)=u

[@hoodedR]

Its 11 pm everyone is sleeping and the vid was on full volume.......... That intro....

[@josealonsoruiznavarro567]

When you are tired, but then you realise that Flammable uploaded a 26 minutes long solution. I felt like Thanos "This put a smile on my face"

[@kabaran2]

I absolutely love this video. This solution was absolutely ingenious , and the way everything simplified so nicely in the end weirdly pleases me. You actually did make my weekend, and as a result I am now a patreon. Keep up the incredible work!

[@Ricocossa1]

I couldn't resist just solving I'' with complex analysis, so in the end it was faster. But it's nice to see those Laplace transforms once in a while.

[@JakeTillman]

I didn’t have a clue what was going on for most of it as I am currently in the 11th year of English secondary school (GCSE year) but you still managed to interest me with your enthusiasm and charisma. Basically what I’m saying is well fucking done and great videos🤙🏼

[@GreenMeansGOF]

Im glad you caught the minus sign error at the end. It was seriously bugging me.

[@heisenbergye1867]

22:17 "But what's our initial condition?" I just suddenly lost myself lol. We've come so far the shit looks nothing like the initial problem lmaoooo

[@AnuragKumar-io2sb]

I don't know these are not taught at University/college ..this is so cool,Feynman technique,Laplace transform ,differential,integral all used up in 1 question

[@amritsahani5268]

Awesome. Seriously, I just came home after giving a test. This legit made my day. This is one of the best question I saw in last few months.

[@user-wu8yq1rb9t]

I love how you love and enjoy Math, Papa. One of the Greatest Video I recently watched, full of great things. Thank you so much my *lovely Papa* ❤️

[@gustavosedano294]

I don't have words to say what i think about this video. Just amazing

[@chizhang3254]

I realized a fact that if I replace t = 1 by t = -1, by observing the expression of I(t) I'm supposed to be getting the same result just because I(t) is an even function of t. But the expression of I(t) that we find out in the end is not an even function of t, but all those steps that we did for finding I(t) do not depend on the sign of t. And one thing that if we replace t by -t in the expression of I'(t) and I'''(t) (it doesn't affect on I''(t)), we will have a differential equation that doesn't have real eigenvalues.

[@copperfield42]

at that triple derivative: and this is to go even further beyond!!! AAAAHHHHHH!!!

[@popescuervin7893]

me two years ago in highschool: wow, this was really hard and cool me first year in college: eAsY

[@azmath2059]

This is f***ing unbelievable. Sheer genius.

[@sugaku9517]

Great result but why did you use Laplace transform at 10:10 when you can just set y(t):=J(t)-pi/2 and obtain y''-4y=0 which solutions have the form y(t)=y(0) * cosh (2t) + y'(0)/2 * sinh (2t) ? (it's quicker and didin't need to know the theory of Laplace Transform ^^ ) (sorry for my bad english I'm a French student)

[@RobertoEmilioRomero]

He did it like that because is his video and that's what he felt like doing. No matter how long or short it took or would of taken he felt like explaining it like that and that's what he did.

[@smrtfasizmu6161]

I mean he could have also used derivative operator and wrote (D^2 - 4)J=pi/2 (D-2)(D+2)J=pi/2 set y = (D+2)J (D-2)y=pi/2 and then solve it that way but as the other person said he likes doing the Laplace transform

[@purcell2319]

Bro, you just got me so hyped to take differential equations. I love your enthusiasm!

[@vkilgore11]

"We're using Laplace transforms!" (Throws chalk) hahahaha

[@setsu2221]

You are actually giving my existence meaning with your videos. Thank you so much. I have been rewatching movies and animes, done maths exercises, and gamed a lot, but it really didn't help. Then I found your channel because of Tibees, and I must say, this definitely is the best cure for boredom.

[@Sugarman96]

Another clean way to do it is to note that the fourier transform of a triangle function from [-2,2] is sin^2(w)/w^2 and the fourier transform of e^(-|t|) is 1/(w^2+1), constants not withstanding. Having noted that, you can use Parseval's Identity to calculate the integral.

[@estebanguerrero682]

Man, that thing inspired me, the way that the result just show up after that parametrization ❤️❤️❤️

[@michaelzumpano7318]

So perfectly explained. So easy to follow. So brilliant. So inspiring. Thank you Pi M and BPRP for this blessing. Possibly the Mozart of Math. Beautiful work boi.

[@吉岡美緒-w3p]

A long and winding load to solve a linear differential equation with Laplace transform. Solve the characteristic equation of the LDE, and build the solution space of it. Then, some exponential functions will appear directly.

[@khaucan5068]

We all know that the cat is the thing KZbin recommended to you and it's what you came for

[@MoeSalamaIbrahim]

I'm watching this now in a weekend nearly two years after the release and this still made my weekend better!

[@peterdriscoll4070]

Complex analysis! Residues! Ha ha ha. But you are on fire.That is crazy.

[@GainsGoblin]

I am a litterature student now.

[@anushreesabnis5856]

I wish I could understand this better, it looks heavenly

[@JamesSarantidis]

I love Differential Equations and Laplace Transforms, they are so useful in Control Systems and Model Simulations. I was totally excited when you transformed the Integration into a initial value problem. Reminds me when I used integration to calculate an area, while my tutor was using an orthogonal approximation and was caught by surprise. The point is I was always afraid of math. Now, with content like yours, I turned my greatest fear into my greatest weapon. Thank you, senpai!

[@HilbertXVI]

'To your newborn son' lol definitely

[@mandlamnisi3816]

This method of using Laplace transform and Differentiating an integral got me so excited! When you threw your chalk, I threw my pen down.

[@helloitsme7553]

This is the best I've ever seen

[@aaronbotterill]

This is my new favourite channel

[@lucasp7630]

I LOVE THIS OMG MY WEEKEND REALLY WAS MADE

[@shashankbalaji4122]

That was an awesome mashup of beautiful pieces of mathematics

[@hopy51]

Nice work! This integral can also be calculated using complex analysis, with the residues theorem. I think it is much easier.

[@HershO.]

I know papa probably won't see this but its wednessday and this still made my weekend. Overcomplicating can make shit easier

[@PapaFlammy69]

[@HershO.]

@@PapaFlammy69 lmao half my comment is rendered moot.

[@JustSimplySilly]

You certainly gave some light to my weekend

[@wanyinleung912]

I discovered this channel today, this is **PURE GOLD**.

[@MichaelJamesActually]

That was pretty awesome. Watching a second time...

[@theopapa8232]

I have another way of solving this differential equation. We have f"(x) = -2π + 4f(x) We add to both sides 2 f'(x) Then multiply be e^(2x) to bith sides and then we get. We actually do inverse of differentiation process (e^(2x) * f'(x) )' =( 2 *e^(2x)*f(x) - e^(2x) * π )' Therefore e^(2x)f'(x) - 2e^(2x)f(x) = -e(2x)*π + c1 We divide by e^(4x) both sides and we get ( f(x) ) ' ( ------ ) =( π/2 * e^(-2x) + ( e^(2x)) c1/4 * e^(-4x) + c2)' Hence f(x) = π/2 + c1/4 *e^(-2x) + c2*e^(2x)

[@SteamPunkLV]

I was so nervous when you made that sign error in the end, I was begging so hard that you notice it and I got so happy when you corrected it and got the correct answer 😂

[@phonon1]

Great video. Really fun example of how solving an integral can be made easier if you can set it up as a diff eq. I solved with annihilators!

[@giovanniz7792]

Wow..I just came back from Germany and this is the first thing I see on KZbin.. lit

[@peterdriscoll4070]

Way out there on the next planet. So many steps. But definitely flammable.

[@kevinfung6697]

Differential Equation,Laplace Transform,Leibniz Rule all in one video!!! Awesome and Amazing!!

[@gammaknife167]

I can like videos, but why can't I favourite it?? I think this is my favourite video I've ever seen. Showing it to literally everyone I know who vaguely is good at maths. NEVER STOP INTEGRATING!

[@KillianDefaoite]

Nice job. Even Wolfram Alpha takes a second before coughing up the answer on this one.

[@PapaFlammy69]

:)

[@unknown360ful]

THE EXCITEMENT IN THE INTRO THO... #NotificationSquadBois EDIT: I checked the list of links in the description... It took some time to pick up my jaw from the floor...

[@unknown360ful]

It's an honor to get a reply and heart from you #HumbleBoi

[@ThePhinista]

This is what I live for

[@sarmilamondal5439]

One of the best integrations i have ever seen.

[@khushalpatil711]

Understood nothing,,,, but i enjoyed

[@PapaFlammy69]

:D

[@joshuabonet]

This made my weekend better!

[@PapaFlammy69]

:)

[@klassjostedt]

Absolutely amazing content, as always! :'D Your excitement when solving these problems really does shine through! A question though: Is it always true that the Dirichlet integral evaluates to pi/2, even with a factor t in the argument of the sine function? Anyways, a beautiful solution!

[@andrei-edward-popa]

If t is posivive, the Dirichlet integral is pi/2, if t is negative, it is -pi/2 :)

[@klassjostedt]

Andrei Popa Oh ok, thanks! :D

[@nathanmacedo2121]

You save me from depression, thank you very much. It's insane

[@benjamilindqvist912]

You know you are excited when you throw your chalk.

[@genuineprofile6400]

Never seen anyone so excited while solving Maths.

[@monadelic]

amazing. your enthusiasm it's amazing

[@studyhelp7479]

Hello! Lovely mathematics, as usual! Given: J''(t) - 4J(t) = -2 pi Could you not simplify the whole process by using the definitions: J(0) = 0 J'(0) = pi as soon as you can? Then you have: L[J] = K L[J''] = s^2 K - s J(0) - J'(0) = s^2 K - pi L[-2 pi] = -2 pi / s => ( s^2 K - pi ) - 4 ( K ) = -2 pi / s => ( s^2 - 4 ) K = -2 pi / s + pi => K = pi / ( s (s+2) ) Partial fractions then gives: K = ( 1/s - 1/(s+2) ) pi/2 And the inverse Laplace Transform gives: J = L^{-1} [ K] J = ( 1 - exp(-2 t) ) pi/2 The rest then follows as shown? CHEERS!

[@Rory626]

Using Laplace transforms here was like cracking a nut with a sledgehammer. By inspection you can see the solutions for J(t) are either cosh(2t), sinh(2t) or exp(2t), exp(-2t) whichever you prefer. Then try a polynomial for the particular integral. Then balance constants. The LTs made me feel a bit sick

[@EngMorvan]

Great example of a solution by Feynman's technique! I got as excited as you in the video and started showing it to the first one I met! I showed it to my cat. He yawned and went back to sleep. 🙄

[@zalden2209]

What a beast ! i didn't totally understand why you can use the partial derivative and the Laplacian transformation but to come up with that man .. chapeau like we say in my country !

[@KarlMarX_93]

You made my day! Amazing integral and approach. Really awesome! 🔥💪

[@silasrodrigues1446]

You did bring more light to my hot brazilian weekend. Congrats!

[@2000vikrom]

The smugness is freaking on another level 😆😆😆😆 I love it even though I understand half of it

[@yogitshankar6348]

Never have I ever seen a Guy this excited over an integral

[@EDoyl]

Monstrous. Absolutely wild. Thank you father.

[@adipocere5487graphomet]

what a thing to watch on the weekend

[@bouteilledargile]

Definitely one of my favorite integrals of all time! Added to my cool math playlist

[@OhDannyBoy512]

Yes! I love your enthusiasm for maths :) That was a very wild ride (I haven’t learnt Laplace) but the ending was still very satisfying! You have a new subscriber~ ^_^

[@Bollibompa]

This technique is so beautiful.

[@gianmarcogotti2874]

Noticing that sin^2 (x) = 1/2*(1-cos (2x)) and that the integral is even you could also use the Cauchy theorem of complex analysis to obtain the same solution....but I naturally know that you had already thought about that! Anyway, congratulations for the brilliant solution!

[@chaos4785]

23:55 so glad u correct it 😂 Btw this is the best video i've ever watched🔥🔥🔥LITTTT !!! Love u🙋❤

[@prabalbaishya6179]

@alysdexia your life is awful