The Hardest Integral I've Ever Done
ฝัง
- เผยแพร่เมื่อ 30 ก.ย. 2024
- 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!
🙏Support me by becoming a channel member!
/ @brithemathguy
#math #brithemathguy #integral
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.
🎓Become a Math Master With My Intro To Proofs Course! (FREE ON TH-cam)
th-cam.com/video/3czgfHULZCs/w-d-xo.html
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
ah yes the direct eta function, I'm still waiting for the mathematicians to release the indirect eta function
😂
😆😆😆
It's too op
Hahaha
A circle be hiding somewhere in here.
🧐
Shall I email Mathologer?
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)
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, th-cam.com/video/l7AErKEE9-4/w-d-xo.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.
"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.
'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
😂
@@BriTheMathGuy come back a few years later with a much better grasp on calculus. off to university for maths soon, thanks for inspiring!
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)_/¯
It really did not disappoint. I would have never expected that "simple" outcome.
Glad you enjoyed it!
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
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....
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
my new way of doing integrals when the teacher asks me to do one on the blackboard: 5:15
Never seen this one before, this is crazy
Right?!
@@BriTheMathGuy yeah?
Damnnn that’s just beautiful, pi really shows up everywhere. Well done
This was some heavy-duty content! Thumbs up
Thanks a lot!
is this another bprp channel now lol
its like the us and uk version of the office except both are good channels
What a compliment!!
This one is truly amazing bro❤️❤️
Very glad you enjoyed it!
@@BriTheMathGuy ❤️...
honestly i think the work (adventure) done in order to get to the answer is just as amazing as the answer itself
why do these ridiculous integrals have an answer related to pi lol
lol I wish I knew!
because pi is a friend of all of calculus.
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
"Isn't it" hmmmm, a reference perhaps
2:50
that escalated quickly. or rather it was already a bomb to begin with.
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)
That was nuts.
Right?
Ok, u did it
Now take the same, but indefinite integral)
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.
After watching this I only had one thought, how am I going to get past algebra?
You can do it!!
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. 👍👍👍👍👍
wow pi/12!!!(factorial!!!(factorial?))
1:26 "I''ll just call our integrand y"
"y who?"
"why wouldyouevendothat"
Can you please provide some stuff on Legendre and Bessel function. Thank you
3:50 why is it ok to divide by x since according to the integral, x can be equal to 0?
I don't think x can be equal to 0, y would be undefined at x = 0
@@ChaoticNeutral6 0:05 ....
@@Rzko I get that he's integrating between 1 and 0 but I feel like lim_x->0 (x^x)^(x^x)... wouldn't converge based on lim_x->0 x^x not converging. The integral would work because the limit from above (lim_x->0+ (x^x)^(x^x)...) would converge. I haven't actually checked that it would but I'm assuming it will be fine as lim_x->0+ (x^x) does
0 is just an edge case, thinking from a rectangle perspective, its just area 0. It doesn’t matter.
That’s such a beautiful problem!! Loved it
It really is! Glad you enjoyed it!
Wow this is crazy man ! Literally took help of all three function to solve ! Great 👍
Glad you liked it!
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)
“….Crazy integral that we got”? Correction: that YOU GOT….you did the intricate work and we did Jack…don’t give this generation anymore false confidence, their egos are inflated enough.
Great job Brian.
What happens if you change the limits to e^-e and e^1/e? :)
Dirichlet is pronounced "durishlay"
SOO much satisfying
I thought so too!
I don't get why Americans never even try to pronounce non-English words correctly. I mean, this is the internet, it's so extremely easy to look it up! Not a slight to you, Brian, it's just something that boggles me.
You're totally right!
so you mean u to the infinite limit of tetration of u...
Hello sir , being a student writing JEE ADVANCED , i would like to hear your advice for cracking the math part .
I’m just waiting to see this as my final exam for Calc II
The answer is pi^2 * (-infinity)
well this was a wild ride...
Me: B.S.E.E. degree. Statistician for a living. Advanced math tutor (engineering level calculus and higher) for more than 20 years. Bring on this integral!
Later me: No clue what the hell I just watched.
@1:08 Pronounced dee-reesh-lay eta function 😅
damn that’s a lotta letters
it would be amazing if you do a video about lambert w function secondary branches
I understood until 0:00
O.M.G... this was awesome !!
Glad you thought so!!
Me watching a normal BritheMathGuy video: I could do that...
Me watching this video: I probably wouldn't be able to do that.
Mr realizing Brian is writing it all mirrored: I COULD NOT DO THAT.
W o w.
Just Wow!
WOW!
That was so much fun!
I HAVE A SILLY QUESTION LIKE WHY IS (X^x) IS PAIRED ON EACH EACH EXPONENT LIKE CAN'T WE WRITE IT AS JUST X TO THE POWER X TO THE POWER X AND SO ON............ OR........ IS IT POSSIBLE TO WRITE IT AS X^(x+1)^(x+1) and so on.......???
This is an integration from 0 to 1 of an infinite power tower of x^x. WHY DOES PI SHOW UP IN THE ANSWER??? What is it about pi that makes it show up in equations that, for all intents and purposes, seem to have NOTHING to do with circles???
Cute substitutions and solution to the problem, but the answer seems to be wrong. Consider the value of the original function at x=0.5, as you add each (x**x) term, surely it gets progressively closer to 0.9999999999...., similarly for any other value of x between 0 and 1. So the integral from 0 to 1 should equal 1
I feel like I'm dumb
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.
That result doesn't correspond with reality.
For all x in (0,1], (x^x)^(x^x)^... = 1. Therefore the area under that curve is 1.
Dirichlet Eta Function is easy to say. Di as in Deez nutz, ri as in REEEEEE, ch like in loch and let as in ath(let)icism. Much easier to say if you're non-English and actually have proper vowels... It's half of a German Surname, based of a French phrase, Le jeune de Richelette The youth from Richelette. And the actual Surname of Peter Gustav is Lejeune Dirichlet. Perhaps Bri, you should learn German, it would help pronounce the names of these German Mathematicians...
you mirrorly write ζ better than I normally write ζ
So puzzled 😕 prof 👨🏫
I would enjoy taking your math class. You're simply gorgeous... sorry to be a creep!!! I guess my sexual frustrations will have to see me learning to solve bizarre integrals in the mean time...
me: wow i understand
my brain: I DO NOT UNDERSTAND ANYTHING IN THIS VIDEO
Although my Wikipedia reference has more trouble getting a Riemann zeta function
A bit late to tell you, should've been here 3 years ago, but Dirichlet is pronounced like Dee reash lay but without the y sound
You can do the integral at 5:07 without using the gamma function. Here is how:
Define I(t) = ∫₀¹ xᵗ dx
Another representation of I(t) follows from evaluating the integral:
I(t) = xᵗ⁺¹/(t+1) |₀¹ = 1/(t+1)
Now find the k-th derivative of I(t) in two different ways:
dᵏ/dᵏt I(t) = dᵏ/dᵏt (t+1)⁻¹ = (-1)ᵏ k! 1/(t+1)ᵏ⁺¹
It's easy to see the pattern and it can be proofed by induction.
Another way to find the k-th derivative of I(t) is to use the integral represention of I(t) and apply the Leibniz rule for integration:
dᵏ/dᵏt I(t) = ∫₀¹∂ᵏ/∂ᵏt xᵗ dx = ∫₀¹ xᵗ (lnx)ᵏ dx
So comparing the derivatives gives us:
∫₀¹ xᵗ (lnx)ᵏ dx = (-1)ᵏ k! 1/(t+1)ᵏ⁺¹
Setting t=k=n-1 gives us the result in the video:
∫₀¹ xⁿ⁻¹ (lnx)ⁿ⁻¹ dx = - (-1/n)ⁿ (n-1)!
Integral over (x^x) ^ (x^x) ^..... where x = 0 - 1 = 1 since from 0 to 1 the function is a close to straight line y=1, after it is going to infinity when x goes from 1 to bigger x.... check excel....
Where the hell did pi come from?! There's no way (x^x)^(x^x)... has a circle in it. Right?
me thinking how he write backwards
Video editing :)
Because he’s flipped the video, we do not see him how we would see him in real life. Any asymmetries in his face are totally backward! Kinda crazy to think about
I don’t understand anything that’s happening here but hey I’m still watching for some reason 😂🤔
Does anyone know what happens when you change the binds of integration to something like 0 to 0.3?
Now explain why pi has something to do with this crazy integral. Where's the hidden circle?
You didn't even try to say Dirichlet correctly. Disappointing.
Why is the the problem at the beginning not Y = x^Y? Why does it have to be Y = (x^x)^Y?
how do you draw in the air is it glass? I'm confused. but a great video ( I understand the maths for sure)!
bro u forgot to say the 2 on top of the pi and 12's 2 cancels eachother so its pi/1 and in romen 1=i so the answer is just 'P'
ur welcm
.
But why is y=(x^x)^y if with the same logic we could have taken y=x^y ?
Where did the -1+n exponent come from in the x ln(x) substitution? 4:52
1:38 "another way to write this would be y equals x to the x all to the y" Is the illegal step he took.
you know your aren't good enough to understand it properly whe you have to pause at the 21st second. And yes i watched the whole video and realised how easy my math now is.
Pov: me as an 8th grader having a stroke seeing how difficult math would get in the future
This is absolutely beautiful
How did you plug it in the lambert Function at the end I am trying to understand for the past 30 minutes
Why was this Recommended to me. I have no clue what I saw or why it is Beautiful. Why is there a Random big e in math
Are you drawing on a mirror or glass or is it editing? I can't seem to wrap my mind around this type of camera view, and I see it a lot.
OMG Hardly to be expected that Pi^2 worms its way out of this.
I havent even got to doing integrals cuz im still in highschool but... This is Interesting, Yes
uses gamma and zeta? that is a nope. Also going to say not possible as the function is not well defined.
*NOICE*.
I fear no function, but that thing... W(x) ... it scares me
Wait... you got a constant from an integral? Or was there more stuff to do?
I graphed y=x^(xy) in desmos and the slope is so large it starts leaning left
Make a program over it ???so we got a general solution 😃😃😀😀😀
The ray of convergency of the serie of W(x) is 1/e, and you integrate from 0 to 1, there is a problem here
And by the way I think it converges to 1 and not pi^2/12
When you said that the answer is beautiful, I knew it would have pi in it.