The Integral of your Dreams (or Nightmares)

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/ @brithemathguy
Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.
#math #brithemathguy #integral

[@BriTheMathGuy]

🎓Become a Math Master With My Intro To Proofs Course! (FREE ON KZbin) kzbin.info/www/bejne/aZTdmJl-irGNedU

[@neiljf1089]

At first I was amazed that he can do backwards writing so neatly. Then realised he just flipped the video

[@HogTieChamp]

I was amazed but then you ruined the magic for me!!

[@manasaprakash7125]

What????

[@offbeatstuff8473]

I was just going to comment the same thing.

[@umershaikh7179]

that is pretty obvious...

[@ParagPardhiNITT]

@@manasaprakash7125 sarcasm dude 😅

[@Ascientistsjourney]

Mathematicians: Look at my integral of my dreams. Physicists: Cool. But does that serve any purpose? Mathematicians: NO, but look at it. It's so magical. ;p

[@123akash121]

truest thing i have heard

[@mathieuaurousseau100]

Next century physicist : hey guys, you will never believe what weird function I'm trying to integrate today

[@jimschneider799]

@@mathieuaurousseau100 - this century's pure mathematics is next century's applied mathematics, because of those meddling physicists.

[@BriTheMathGuy]

😂So True!

[@Ascientistsjourney]

@@BriTheMathGuy woah you saw my comment. Thanks bro you made my day 😊

[@tnk4me4]

Never have I understood "Sufficiently advanced math is indistinguishable from magic" more than this very moment.

[@GreenCaulerpa]

Except the original quote was “Any sufficiently advanced technology is indistinguishable from magic” from Arthur C. Clarke‘s book „Profiles of the Future: An Inquiry into the Limits of the Possible“ (1962). But I agree this integral is pretty much nightmare stuff if you haven‘t seen once how to solve it.

[@tnk4me4]

@@GreenCaulerpa Yes thank you for explaining the joke. You get an internet cookie. Congratulations.

[@GreenCaulerpa]

@@tnk4me4 yummy, thanks for that cookie!

[@rmxevbio5889]

@@GreenCaulerpa nice quote!

[@az0rs]

Holy cow that’s the prettiest integral I have ever seen

[@BriTheMathGuy]

I think so too!

[@mathe.dominio4765]

👌

[@turbostar101]

And he’s doing it backwards!

[@eduferreyraok]

I would took a little twist over the improper integral, by applying a laplace transform which matches with the definition : F(s) = L { f(t) } = integral from 0 to inf of f(t). e^(-st) dt .

[@mrnogot4251]

2:40 dude nice thank you for being aware that you can’t just interchange infinite sums and integrals willy nilly.

[@HeinrichHartmann]

He did not give an argument, though. He just mentioned "uniform convergence". But why would this sum converge uniformly? ln(x) has a singularity at 0, so I am not sure about uniform convergance on [0,1].

[@grekiki]

@@HeinrichHartmann Series for e^x converges absolutely

[@markusdemedeiros8513]

​@@HeinrichHartmann I can try to fill in the details for anyone interested: x log(x) is bounded on (0,1]: I will not do this here but it is concave up, has a minimum, and the limit at both 0 and 1 is 0. Therefore there's some closed interval containing all values of x log x for x in (0,1]. The power series of e^x converges uniformly on any closed subinterval of it's interval of convergence R, so the series for e^(x log x) converges uniformly for x in (0,1].

[@holomurphy22]

@@markusdemedeiros8513 One could just say that x log(x) is continuous on (0,1] and can be extended continuously to [0,1] as it converges to 0 in 0. The extended function is bounded because of 'extreme value theorem' and thus x log(x) is bounded on (0,1] I may be misspelling things a bit

[@onradioactivewaves]

@@markusdemedeiros8513 thanks, I appreciate that summary.

[@ilyaxi]

What's most fascinating is the way he looks to be writing from right to left for us. It's surely inverted but stil.. Thanks for the vid

[@cnvrgnt]

That was NOT the result I was expecting form this. Absolutely beautiful

[@BriTheMathGuy]

Glad you enjoyed it!

[@kaasmeester5903]

It is. But I still hate integrals :) I never had much issues with other mathematics (up to a masters in EE) but integrals always turn into these crappy little puzzles that apparently I'm just to dumb to solve.

[@Roboboy-v6]

As an engineering student my first instinct was to use a euler's method of approximation cause "fuck that work" LOL

[@adamuhaddadi5332]

stupid approximateurs >:(

[@bowenjudd1028]

It’s ancient, but it works

[@chungus478]

You know you're an engineer when using π=3 does not seem like an approximation

[@bowenjudd1028]

@@chungus478, and a mathematics or physics student if it does.

[@lucidmath5481]

we need more integrals like this, this is amazing

[@sebastienruhlmann3917]

The actually important explanation for interchanging sum and integral is brushed away like nothing. This took away the beauty of it.

[@sourabhparadeshi4162]

I have my term exams in few days and watching this is satisfying ❤️

[@tamazimuqeria6496]

Same here, good luck

[@sourabhparadeshi4162]

@@tamazimuqeria6496 good luck

[@BriTheMathGuy]

Best of luck all!!

[@heh2393]

How was it?

[@tommassspunis8184]

Damn i got stuck watching this video and the integral of e^-x^2 in loop because at the end of each video the guy says “click on the video on the screen” and its an infinite loop :D

[@BriTheMathGuy]

You've fallen into my trap!!

[@Muhahahahaz]

Oh no… I actually just arrived at this video from a different video, but I could end up in the same loop as well Next step: make sure that every sequence of video links eventually leads to this specific loop. Reminds me of the Collatz Conjecture… 🤔

[@joshuaisemperor]

blew my mind. Never seen summation and integrals after each other.

[@BriTheMathGuy]

Pretty cool right?

[@joshuaisemperor]

@@BriTheMathGuy yeah but it also feels intimidating for someone who still has to pass his Calc 2.

[@BriTheMathGuy]

You can do it though!

[@engr.rimarc.liguan1795]

This was the cutest introduction of solution I have ever seen in addition to the handsomeness of the one who introduced it. 😅🤭 Bravo!

[@FatihKarakurt]

Glass pane works really well. If you can dim the lights over your hand it will be much better.

[@이름-x6s]

I am a university student in Korea. I was always interested in math, and I happened to see your KZbin while I was looking for a related KZbin while preparing for a math test. I think there are a lot of fun and informative contents. I hope your KZbin will be better and I will continue to look for it often. Thank you!

[@limagabriel7]

do u guys learn calculus in high school in korea?

[@이름-x6s]

@@limagabriel7 Yes, I do learn, but for example, in the case of calculus that utilizes two or more variables, I learn properly in college.

[@uggupuggu]

Why are you named Apple Boss

[@adb012]

Something that surprised me more than the continuous sum being equal to the discrete sum is the bounds of those sums. The continuous sum of x^(-x) from 0 to 1 equals the discrete sum of n^(-n) from 1 to infinity... *SAY WHAT?!?!?*

[@Francesco-bf8cb]

I'm here to comment just to make your video more popular

[@BriTheMathGuy]

Thanks so much!

[@God-ld6ll]

maybe more like a sophomore's nightmare to some i'd imagine

[@BriTheMathGuy]

😂

[@jackweslycamacho8982]

It's even crazier how fast it converges. For the first 7 values of n you literally have n digits of precision, after that it the rate of precision keeps getting higher.

[@captainhd9741]

Care to share an example? I am admittedly too lazy to figure out the value of the sum and how fast it gets to these values.

[@jackweslycamacho8982]

@@captainhd9741 use desmos and input sum for sum and int for integral

[@captainhd9741]

@@jackweslycamacho8982 I prefer Wolfram but good idea!

[@MarioRossi-sh4uk]

@@captainhd9741 1 1 2 1.25 3 1.28703703703704 4 1.29094328703704 5 1.29126328703704 6 1.29128472050754 7 1.29128593477322 8 1.29128599437787 9 1.29128599695904 10 1.29128599705904 11 1.29128599706255 12 1.29128599706266 13 1.29128599706266

[@thisisnotmyrealname628]

7:08 moment of satisfaction

[@sjzara]

What I don’t understand is how mathematicians make such amazingly leaps such as the various substitutions to get to the answer.

[@braedenbertz1063]

Its a lot of trial and error, looking at past results and seeing if there are parallels, and a lot of luck :)

[@sciencewithali4916]

I am genuinely getting addicted to your videos !

[@BriTheMathGuy]

Glad you like them!

[@carljohanr]

Really nice results - I assume there is no closed form for the sum, but it made me a bit surprised at the end that you never touched on that topic.

[@assasin1992m]

There is, it equals sin(pi) / gamma(pi/2)

[@captainhd9741]

@@assasin1992m What is sine doing here? 🤔

[@captainhd9741]

@@assasin1992m makes me wonder if there is a complex extension for z^(-z) integral

[@ha14mu]

Isn't sin(pi) 0?

[@assasin1992m]

@@ha14mu yes, but the limit toward pi in this expression converges to a non zero result

[@ankitbasera8470]

I really admire the way you explain, not in a hurry

[@JayTemple]

I love the fact that a video about calculus was interrupted by an ad that talks about partials (dentiures).

[@BriTheMathGuy]

😂

[@ejb7969]

That's because calculus is a subject you can really sink your teeth into! And if anyone is thinking "That joke really bites", I beat you to it. Chew on that one!

[@noone8253]

Got a similar problem in a calc 2 exam, I was very confused and thought it was unsolvable, still processing how to get a numerical value for the solution, very nice video!

[@marshian__mallow2624]

For an integral like that. You don’t get a numerical value

[@pvshka]

Friggin high school maths still giving me headache. Good job

[@grinreaperoftrolls7528]

I freakin love calculus. I thought this was gonna be really scary at first.

[@lukekolodziej9631]

I honestly think I'm more impressed by how good you are at writing backwards. LOL! Good video

[@destructiveodst1199]

He’s not writing backwards it’s just mirrored lol

[@Unifrog_]

I'm impressed by how well he can write mirrored then /jk

[@Chrisuan]

Love your content! You can really feel your love for the math

[@BriTheMathGuy]

Glad you enjoy it!

[@K_V-S]

*We can keep going on exploring & doing maths .. cuz it only demands three qualities of our mind* 1. *Curiosity to know* 2. *Using only knowledge i.e. No belief system* 3. (most important) *Focused mind to dig deep into the question*

[@HaLKer5]

Wow, this was much better than i expected! Truly beautiful!

[@ThomasHaberkorn]

Omg the twist at the end is quite a shocker

[@sauravrao234]

I think what is amazing is that the integral of x^x within the same limits gives the same summation but with a (-1)^n, hence having alternating plus and minsu. So the integral of this video outputs a greater value than integral of x^x within the same limits, which makes sense. Because x^-x is bigger than x^x in this interval of 0 to 1.

[@FernandoRuiz-rf1om]

Does the final infinite sum converge? Awesome integral btw!

[@BriTheMathGuy]

Thanks! and yes it most certainly does! (around 1.29 or so)

[@sophiophile]

@@BriTheMathGuy is there an exact identity for what it converges to, or did you just get this by approximation?

[@leofisher1280]

@@sophiophile there is no closed form for it sadly so all you can do is solve it numerically.

[@davidgillies620]

The good news is the convergence is extremely rapid. The first ten terms of the sum give you the value of the integral to about 3 parts in a trillion.

[@olbluelips]

@@tBagley43 almost all this kind of stuff has no closed form

[@muqeetsoheb6708]

Its intresting how he uses just SMALL PART of BOARD to explain such complex problems whereas for our teacher need two full boards

[@Pod_TM]

Uniforme convergence isn't the reason you can do the important early swap sum integral, the hypotesis are : if we note u_n to be the function inside the sum (here x^n/n!) Then we can use the theorem under the conditions that sum(u_n) converges (i believe not even necessarly uniformly), integral(u_n) converges and sum(integral(absolute value(u_n))) converges. Not a lot of these has to do with uniforme convergence

[@Rkcuddles]

A continuous sum becomes a discrete sum. Totally wish you extended the video by 1 minute to really nail that in for the younger audience that may be casually watching this fantastic puzzle

[@paolomeola6180]

Just do the transform y = -x and then you solve in dy! we already know the answer!

[@arthurkassis]

I'm in the sophomore year so I understand anything when start caculus, but I still loving your content, Ive always been ahead of the current math subject of my school so I tjink that watchint this will also help a bit more. For now I'm studying analytical geometry, is easy and I like, and calculus I'll some time soon

[@terminusfinity009]

this makes me want to take out my scientific calculator

[@ooflespoofle3691]

"He will never cancel the n!" *spits out cereal*

[@alperenerol1852]

I was gonna discretize the domain and calculate the area by numerical methods.

[@mrgadget1485]

That was beautiful - and scary!

[@rachit7645]

Wolphram Alpha says the final sum is approximately: 1.2912859970626635404072825905956005414986193682745223173100024451369445387652344555588170411294297089849950709248154305484104874192848641975791635559479136964969741568780207997291779482730090256492305507209666381284670120536857459787030012778941292882535517702223833753193457492599677796483008495491110669649755010519757429116210970215616695328976892427890058093908147880940367993055895352006337161104650946386068088649986065310218534124791597373052710686824652246770336860469870234201965831431339687388172956893553685179852142066626416543806122456994096635604388523996938130448401015323385569895478992261465970681807533429122890910049951364103584723741679660994037428872280908239472403012423375069665874314768350298347009659693019807122059415474239188849548892043147840373896935928327449373018601817579524681909135596506205768427008907326547137233834847185623248044173423385652705113744822086069838116970644789631554803110868684680780701057034230000954776628299270222642661822130291609344850492556799951212817650810621807347685511270748919272166418829000073661836619726956875357964537813752368262924072016883803114377731170

[@robertmonroe9728]

Try to integrate it to infinity. Integral converges. But this way will not work

[@justinkane290]

It's like turning a jig saw puzzle into a Rubik's cube.

[@SQRTime]

Hi Justin, if interested in math competitions, please consider Finding Sum of Digits kzbin.info/www/bejne/gaiadJaCmKiIm5Y and other videos in the Olympiad playlist.

[@akankshasharma7498]

Man! You love Gamma function so much 🤣🤣🤣🤣🤣

[@BriTheMathGuy]

Yes I do!

[@adammohamed5256]

Well done! This is really amazzzing !

[@colinslant]

That is a very remarkable and beautiful result.

[@bowenjudd1028]

I actually lost all sense realizing how cruel sone equations can be.

[@elmogus572]

This channel is amazing !!!!!!

[@abhaysharma3467]

i'm in class 12 but i can understand this😍

[@Thechinkills]

dude i graduated with my engineering degree why am I still watching Math videos? beautiful vid btw

[@kqp1998gyy]

An effective channel. Thank you

[@BriTheMathGuy]

Glad you think so!

[@xxbananahanahxx3012]

I like your funny words, magic man.

[@ThomasKundera]

I would be unable to do it by myself without guidance But the whole video was a beautiful journey where I was smiling at each new trick Just disappointed it didn't arrived to some usual function development

[@nolanrata7537]

A solution that doesn't require substitutions or knowing the gamma function is to integrate (-ln x)^n*x^n between 0 and 1 by parts n times to find that it is n!/(n+1)^(n+1) and the final results comes naturally.

[@divisix024]

Well that’s just how you prove the gamma function evaluated at n+1 is the same as n!

[@josephhobbs4680]

approximately 1.29

[@Abel-Ramanujan]

You made it so simple :)

[@BriTheMathGuy]

Glad you think so!

[@joaquingutierrez3072]

Amazing video!!

[@Leeanne750]

Good explanation!

[@BriTheMathGuy]

Glad you think so!

[@AhsanAli-gu4bm]

we can easily solve it by taking natural log and apply integratiom by parts

[@rbrowne2998]

Extraordinary! I didn't see it coming.

[@rdsmofficial]

I have no idea what is going on and im thinking of studying applied maths....

[@mastershooter64]

damn that's fuckin sick dude!

[@AnakinSkywalker-zq6lm]

This might help me with a problem I’m working on

[@AliceOutlandish]

I don't know anything about calculus(only took precal my senior year of highschool) and this is scaring me and I have no idea what's going om

[@HershO.]

Dude I checked on wolfram alpha and the sum is = 1.29129. See this is so cool cuz 129 is repeated. I love this.

[@Timmmmartin]

Equal to 430/333 if 129 were to repeat indefinitely.

[@HershO.]

@@Timmmmartin no dude this is its exact value, its not recurring. It's not particularly fascinating of a fact but its cool.

[@Timmmmartin]

@@HershO. Do you have the wolfram link please?

[@MaximQuantum]

Nightmare, bc the solution can only be solved with an approximation 😔😔😔

[@penta4568]

As someone who has graduated with an Applied Mathematics degree, I probably should know what’s going on here😅 (I do but it’s still a headache, I made it out & enjoy the fact that this stuff doesn’t come up often irl)

[@darkarchon2841]

Shit, this is much more interesting than when I was learning that in uni...

[@DebashishGhoshOfficial]

Sure, if you show it like this. But you can easily break this integral into Riemann sums and you will arrive at the same expression. And that is very normal.

[@ronaldronald8819]

Could never work that out myself but it fun to look at.

[@ThomasHaberkorn]

wow pretty nice. Did Max Plank do something similar when he figured out that energy is quantized? (i.e. using a discrete sum vs. an integral)

[@regulusxxz]

Great thinking

[@wallstreetoneil]

Sort of yes & sort of no. Plan(c)k was presented with an actual data set (the ultraviolet catastrophe) that showed that the energy didn't explode off to Infinity (like in a 1/x graph as x approaches 0 and as predicted by Classical Physics - i.e. as the wavelengths got smaller), but it turned back down and headed towards zero like a Log Normal Distribution. So the data was real - and in his own words "out of desperation" he was willing to try anything to figure it out and then hope in the future someone could explain the physics of WHY? (which Einstein did and was given the Nobel Prize for - Photoelectric Effect). So he tried many things, and one of those things was to assume that if you looked microscopically close enough to a 1/x graph, it wasn't actually smooth/continuous - but it was actually discrete in tiny, tiny, tiny increments. It took a long, long time of incredible focus, but that natural conclusion to this assumption resulted in an equation where the denominator in the equation had a term that increased faster than a similar term in the numerator and it pulled the resulting curve, like a Log Normal Dist back towards zero. So, "out of desperation", because he thought this was incredibly important for science, he 'hacked' a problem, and came up with an answer, from an assumption that at the time made no sense, but he had the Mathematical skills and dedication to find a formula that explained what the data showed. He's a legend

[@ThomasHaberkorn]

@@wallstreetoneil agreed. Planck totally hacked maths to result a log normal distribution. I'm curious how long he worked on this problem and on his failed attempts

[@wallstreetoneil]

@@ThomasHaberkorn 6 years of intense focus on only this problem is my understanding - similar in a way to Andrew Wiles Fermat's proof which he basically did every night for 7 years after returning home from work - and then of course there's Perelman and his Poincare proof and declining the Field's Medal which takes it to a whole other level. According to Russian Mathematicians, Perelman has completely retreated from the world as is working nonstop on the Navier-Stokes Existence and Smoothness problem - which is one of the 7 most important open problems in Mathematics. It wouldn't be a huge step from all the Ricci Flow stuff he used to prove Poincare - and if he accomplishes this, he will undoubtedly go down in history as one of the true greats by solving two of the greatest Math problems of the 20th century.

[@ashutoshkumarjha41]

Love the way you speak and write.

[@BriTheMathGuy]

Thanks very much and thanks for watching!

[@guestmode867]

kzbin.info/www/bejne/onnMZmaHmteYfqM 3:18 Actually, from here: we can pull out (-1)^n/n! outside the integral and the remaining integral from 0 to 1: ∫(xlnx)^ndx becomes a variant of the gamma function: -(-1/(n+1))^(n+1) gamma(n+1) Gamma of n + 1 is also n! so the final result is: summation from 0 to ∞[ {(-1)^n/n!} * n! * -1 * (-1)^n+1 * (1/n+1)^n+1 ] n factorials cancel out and the exponents of 1 are added up: ...[(-1)^(2n + 2) * (1/n+1)^(n+1)] Since the exponent of -1 is always even as we are taking a discrete sum of whole numbers, it is always positive 1 so we can remove it. = summation from 0 to ∞ of (1/n+1)^(n+1) changing the bounds of the summation by +1 and subtracting 1 from the n terms we get: summation from 1 to ∞ of (1/n)^n Since 1/n = n^-1 Answer = summation from 1 to ∞ of n^(-n)

[@Mkvyas1]

Whole video like... Question. Which letter is next after letter A in alphabet ? Answer. Which letter is before C in alphabet ?

[@wtomalik]

And this is why I didn't continue further in mathematics...

[@PrinceKumar-og8kl]

what a beauty!

[@Sahu_-pk6op]

Just another question for IIT-JEE Aspirants.

[@DJ_Force]

That's one of those test questions where you are convinced you got lost and got the answer wrong, even if you get it right.

[@firemayro]

...oh dear i just got into this unit.

[@aberattedaniketdatta4126]

Please upload videos on IMO problems too they are also very deep

[@logantotesrocks1507]

He’s like the Bob Ross of mathematics

[@rabeakhatun2819]

Just wow 🔥🔥🔥🔥

[@xeerakazhar9023]

How do you write laterally invertedly by the left hand?

[@BriTheMathGuy]

Video editing :)

[@felipeyoshino6951]

Kid: Hold on, dad! Look at this little integral I've found! Dad: Stay away from that, it's just a trap that will make all of us to lose our way home just like those skeletons.

[@thedra9ongod]

5:37 me: WHERE THE HECK ARE WE GETTING

[@ZeDlinG67]

now please do the integral of x^x in -1 to 0 :D

[@fredericoamigo]

Awesome vid! Good job!

[@BriTheMathGuy]

Thanks for the visit!

[@DrJens-pn5qk]

Very nice. But what is the value of the final sum? Does it even converge?

[@lobsterfork]

1^-1 + 2^-2 + 3^-3 + 4^-4 + 5^-5 + 6^-6 = 1.29128472051. I think more terms are necessary, for precisions sake, but you get the main idea.

[@-10ZetMazur01-gc2ym]

isnt a definite integral supposed to have a numerical value? It is not clear to me how it became an infiinite sum. Is that result with the infinite sum convergent or divergent?

[@sonarbangla8711]

Extremely interesting indeed.!!!!!

[@BriTheMathGuy]

Glad you think so!

[@sonarbangla8711]

@@BriTheMathGuy I wonder if these functions need to be analytic or converge?

[@Total_Syntheses]

sophomore's dream..

[@padmasangale8194]

How he interchanged the summation and integral signs at 2:30 please someone help me😢(btw i am class 11th student and jee aspirant)

[@saxbend]

But do we have an expression for n in terms of x? If not all we have got is the equivalent of defining a new variable to be a a function of the solution and then declaring that we have a solution by presenting it in terms of that variable.

[@ericsimonetti5876]

from how this plays out it seems to me that the answer is not dependent on x at all, meaning the answer is that summation for all real x (in the domain of x^-x of course). This fact seems pretty amazing to me on its own but given the fact he didn't mention it I might be wrong? If anyone knows for sure I'd love to know

[@KidNapPingNo1]

Would be interesting if the series which resulted from this integral converges to some value :) next challenge ? ;)