The Hardest Integral I've Ever Done
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- Опубликовано: 25 апр 2021
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I'm not joking, this is one of those hard integral problems. It's time to tackle one of the hardest integral ever - that I've computed at least :)
I hope you're enjoying these hard integral questions and hard integral problems with solutions!
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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.
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could I expect to see this level of complexity for integrals in calc II or would this be more advanced content?
@@cuad0130Calc II won’t deal with this. You will be calculating much simpler integrals. Don’t relax too much though, Calc II is a weed-out class
"This is completely legal" is exactly what a criminal would say.
👮♂️🕵️♀️👨⚖️
no comments??
@@mhm6421 You mean a statement?
Bored student: “Prof? I wanna see an integral with all my favourite special functions!!”
Professor: “Say no more, fam!”
😂
There’s no need to call me fam, professor.
BPRP be like:
I don’t even know these functions and I have a 90 something in calc 3
@@nabranestwistypuzzler7019mainly cuz they're not much involved in calc 3 unless you're knee deep in pure mathematics. But they're very popular functions within the pure maths community for many reasons but mainly because of the reimann hypothesis (for the zeta function), negative/fractional factorials (gamma function) and obviously, inverse of a function where the X is in both the exponents and base (w function). They're satisfying I'm their own way
A circle be hiding somewhere in here.
🧐
Shall I email Mathologer?
ah yes the direct eta function, I'm still waiting for the mathematicians to release the indirect eta function
😂
😆😆😆
It's too op
Hahaha
Spoiler alert: The answer will not disappoint you. I recommend watching it to the very end.
Glad you thought so!!
Can confirm, was not disappointed!
Thanks now I didn’t waste time not getting bored
I heard riemann zeta function so i knew that there would be π in the awnser. (Γ(1/2) is also √(π) so that could've been a candidate aswell)
It really did not disappoint. I would have never expected that "simple" outcome.
Glad you enjoyed it!
"Dirish Lay"
Isn't it more like "Direesh Lay" ??
@@gamedepths4792 yep
I knew I couldn't do it 😭
@@gamedepths4792 isn't it the same, sounds like a french name so it would be "dirish leh" not "lay"
He was German and pronounced it “Dee-ree-kleh”. That’s also how every math prof I ever had (and mentioned him) did it.
Many of you may be wondering, "how does Brian know that the tower converges on the interval [0, 1]?" Let me help answer this question. In this other video that Brian made a while ago, ruclips.net/video/l7AErKEE9-4/видео.html, I wrote in the comments how the power tower, with base x, converges when x lies in the interval [1/e^e, e^(1/e)]. In this video, though, the base of the power tower is x^x, rather than x. The minimum value of x^x is (1/e)^(1/e), which occurs at x = 1/e. Because 1/e < 1, (1/e)^e = 1/e^e < 1/e < (1/e)^(1/e). Additionally, 1 < e^(1/e). So x^x takes on the values of the interval [(1/e)^(1/e), 1] when x lies on the interval [0, 1], and since [0, 1] is contained in the interval [1/e^e, e^(1/e)], the integrand actually converges to the explicit expression for the given interval of integration. The power series is also guaranteed to converge for the same reason, since the convergence of the power series for W is interdependent on the convergence of the power tower.
So in fact, every manipulation Brian used to evaluate the integral is valid, and the integral is indeed well-defined, because the integrand is well-defined on the interval of integration.
While the exchange of the series and integral is valid, he never actually justified it.
@@jadegrace1312 I know I hop in a bit late but that is exactly the "issue". I understand that, during the proof, he didn't want to lose time with justifing, every time he has to compose an expression with any function, that the expression is on the rigth domain to be composed by the function (when he composes with ln or W for example). But the permutation of the itegral sign and the series isn't obvious at all, and I think it is a real issue as, today ,in most math proofs, thoses permutations are less and less explained, while they are the main difficultie of the demonstration.
I believe that here, the "easier" way to proove that we can permutate both the integral and the series, is by showing that the integrated function is the main term of a uniformaly convergent series upon the segment [0, 1].
Otherwise, it was a very cool video, thanks a lot for the proof and keep going with the amazing work you've been doing.
PS : sorry if there are any wrong terms used in my commentary, I'm french and so I'm not used of using english mathematicals terms in writting.
And swapping the integral with the Σ at 5:00 is justified by Fubini?
@@jadegrace1312 I never said he did justify it. Why are you strawmanning me?
@@lofro327 My comment was with regards to the convergence of the function on the interval of integration, not with regards to the exchange of the integral and summation. The latter does not even need context, since it is well-known that Fubini's theorem (or, more precisely, Tonelli's theorem) justifies it.
This is like the final boss for integrals, you need your best equipment to conquer it.
'Im challenging you to make it through the video'
Me who doesn't know any calculus and watches as if its in another language:
Easy
😂
Never seen this one before, this is crazy
Right?!
@@BriTheMathGuy yeah?
Damnnn that’s just beautiful, pi really shows up everywhere. Well done
honestly i think the work (adventure) done in order to get to the answer is just as amazing as the answer itself
Been watching your calculus videos for a long time, you explain things very well. I learned a lot of calculus, just by watching your content. I hope that you continue to make more of them, I write down all the examples you do. 👍👍👍👍👍
It took me longer than I care to admit, but... I managed to derive the integral at 5:10 by making the substitution t = - (n+1) ln(x). This make the integrand (-1/(n+1))^(n+1) t^n e^(-t) dt, and the bounds of integration are from infinity to zero. That substitution didn't show up out of thin air, by the way. I got there by first trying t = ln(x) (which gave the exponent the wrong sign), then t = x ln(x) (which required the Lambert W function just to rewrite the integrand), and then t = - ln(x). With that last one, I wound up with the integrand (-1) t^n e^(- t (n+1)) dt, so I first tried t = - ln(x)/(n+1), and then realized I needed t = - (n+1) ln(x).
I added that rather long-winded explanation to illustrate that sometimes, you need to try several things when solving a problem.
i don't even know you, but i'm so proud of you
What kind of match Psycho you are
What has this crazy integral got to do with circles? 🤔 If there is anything 3Blue1Brown taught me it’s that whenever there is pi in your answer there is a crazy link to circles (although it would be sad if many cases have links to circles which go unnoticed due to cancellation like pi/pi).
try complex analysis
Lol yeah, pi/pi circle go brr :v
Evaluating zeta(2) is just the Basel problem. 3Blue1Brown has conveniently actually made a video explaining why pi shows up there. As for the original problem, I couldn't say.
@@fahrenheit2101 Just go backwards, take an interpretation of the basel problem's circle and just perform all calculations backwawrd
Perhaps the result is the area of a circle with radius = sqrt(pi / 12). ¯\_(ツ)_/¯
my new way of doing integrals when the teacher asks me to do one on the blackboard: 5:15
This was some heavy-duty content! Thumbs up
Thanks a lot!
That’s such a beautiful problem!! Loved it
It really is! Glad you enjoyed it!
What are *your* favorite/crazy integrals?!
integral from -1 to 1 of 1/x
Gaussian integral is my favorite but my favorite derivative is differntiation of e^e^e^e^e^e^e^x. It's answer is amazing.
Integral of x *3*3*x........ From 0 to 4
It does not make sense
Damn you're so good....
Your videos are really awesome! I've never enjoyed Maths until I started watching your videos keep up the good work 👏
Wow that's Awesome! Thank you so much!
Sorry to pick up a random conversation , but does that mean that you haven't watched 3Blue1Brown channel?! ;-)
@@informationparadox387 Nope. Never heard of it.
@@borsalinokizaru7382 that's crazy
So much information in so little time! Thank you sir
My pleasure!
I changed my field of study 3 years ago. Im now a UX designer yet i’m still fan of calculus since it was the only talent i had and hated every thing else in engineering school.
Wow this is crazy man ! Literally took help of all three function to solve ! Great 👍
Glad you liked it!
That was nuts.
Right?
This one is truly amazing bro❤️❤️
Very glad you enjoyed it!
@@BriTheMathGuy ❤️...
that escalated quickly. or rather it was already a bomb to begin with.
W o w.
Just Wow!
WOW!
That was so much fun!
Cool! Definitely spreading this.
The editing on this video is superb!
Thank you so much!
Love your explanation.
So glad!
That's insane!! What a great integral.
that's crazy good, thanks for the video
Thanks for watching!
I can’t believe the RUclips algorithm took this long to recommend you to me. I love your channel! Thanks for the content.
My mind blew away. Need to make it once more.
I’m just waiting to see this as my final exam for Calc II
it would be amazing if you do a video about lambert w function secondary branches
The hardest integral I’ve ever done: 7 minutes
It takes me 7 minutes to find the integral of e^x 😭
Dam you must be pretty new to this
@Hari Venkataraman its a fkn joke dude ^^
@@Daily_Questions820 that's the dumbest joke.. at least if he said partial integration it would've made sense..
@@pianoforte17xx48 u're thinking 2 steps ahead, so sure missing the point on step 1, not all jokes are meant to be world class xd
@@Daily_Questions820 lol I guess
4:56 Dominated convergence theorem: “Am i a joke to you?”
Bro, who verifies the applicability of theorems?
Edit: Out of my own volition, and not because my teacher threatened me, I feel the need to say that verifying a theorem's conditions is very important (Send help please)
Just wow, thats so fucking awesome
You're good. Keep it up !
"Isn't it" hmmmm, a reference perhaps
2:50
Fun to watch, not so fun to solve - but an elegant solution is the reward!
O.M.G... this was awesome !!
Glad you thought so!!
well this was a wild ride...
That was awesome!
Glad you liked it!
I actually visited your site and i wanted to know under which course i will be able to learn about advanced integration like you have shown in the video and under which category it comes under? because i am really curious to learn about new functions. btw love your content and ya i made it through the video :) .
Great one!
Thank you! Cheers!
Amazing video!!
Glad you liked it!
Can you please provide some stuff on Legendre and Bessel function. Thank you
This is absolutely beautiful
Greatly enjoyed this one ...
Glad to hear it!
Wow that is impressive!!!
Amazing!!
You are!
1:26 "I''ll just call our integrand y"
"y who?"
"why wouldyouevendothat"
Hello sir , being a student writing JEE ADVANCED , i would like to hear your advice for cracking the math part .
What an interesting problem and what an interesting solution.!!!!!
It involved the Basal Problem as a bonus too.
how do you draw in the air is it glass? I'm confused. but a great video ( I understand the maths for sure)!
Magical
Excellent
you mirrorly write ζ better than I normally write ζ
Thank you for using “ln” to denote natural log
SOO much satisfying
I thought so too!
You look so proud to have solved this yourself
Crazy answer
Do you have a Gaussian integral type representation for x^^x? (here ^^ is the tetration symbol, x should be real and positive) If yes, what is the derivative of x^^x orthe integral of x^^x in a positive interval? (e.g. from 0 to a, a>0)
What happens if you change the limits to e^-e and e^1/e? :)
Awesome!
Thank you! Cheers!
I am not very aware of those functions cuz i am school student, but i have seen them and understand a bit about them, so yeah i got through the video. I think i now need to figure out what is lambert function and etc :)
My teacher taught us about gamma function but i think i need to figure out more :D
Thnx
Also, π²/12 is so beautiful answer 😂
How did you plug it in the lambert Function at the end I am trying to understand for the past 30 minutes
This is why I love math
You are so good at integration
How much time should I spend on solving algebraic problems to write new stuff by wave of the hand?
Awesome bro
Thanks!
Is there any conditiom that can change the order of integral and infinite sum?
1:45 hold on for a moment, if we do that we first have to know if the limit converges, how do we know that over x in {0,1} that it converges?
4:50 what about the other branches for Lambert W? I can barely find anything on the internet discussing them nor their series representations
yeah I trust you
I kept clicking on the videos you recommend at the end and now im stuck in a loop, pls help
Lol it happened to me once
I like your videos !
Thanks! Very glad to hear it!
This is a beautiful solution! I can't say I completely understood it, but the result is so surprising. By the way, I wonder what's the average age of your viewers, so if someone here in the comments has any idea, I'll be glad to hear.
I am not really that good at math but I just had a feeling the integral probably had something to do with pi, and I was amazed when I saw the final answer!
How did you plug in the lambert ?
My mind is blown
Does anyone know what happens when you change the binds of integration to something like 0 to 0.3?
Best problème ever
3:56 alternate: w(w(x)e^w(x))= w(x) on the other side, when you take w(x), w(x) = w(x). Solved!
Let’s just appreciate how neat his mirror image handwriting is
So puzzled 😕 prof 👨🏫
After watching this I only had one thought, how am I going to get past algebra?
You can do it!!
I graphed y=x^(xy) in desmos and the slope is so large it starts leaning left
very nice, not as demandjng as i thought it would be
Glad you enjoyed it! Maybe I'll have to find something harder 🤔
Where did the -1+n exponent come from in the x ln(x) substitution? 4:52
Can you graph the original integrand?
When you said that the answer is beautiful, I knew it would have pi in it.
I guessed that pi would be in this
but can you actually switch integral with infinite sum?
Is it infinity as soon as we integrate further than 1?
Great!!!!!👍👍👍👍👍👍
無理ゲーと思えるところから、いつもの無限テトレーション→W関数で、
最後にガンマ関数でスッキリしちゃうとは
これは妙味ありまくりですねえ～～～
Glad you liked it! (I don't know what the rest of your comment say's though)
@@BriTheMathGuy I would just copy into Google Translate, lol. My limited katakana knowledge shows that he's mentioning tetration, though.