The Hardest Integral I've Ever Done
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- Äas pĆidĂĄn 25. 04. 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 wonât deal with this. You will be calculating much simpler integrals. Donât relax too much though, Calc II is a weed-out class
"This is completely legal" is exactly what a criminal would say.
đźââïžđ”ïžââïžđšââïž
no comments??
@@mhm6421 You mean a statement?
Bored student: âProf? I wanna see an integral with all my favourite special functions!!â
Professor: âSay no more, fam!â
đ
Thereâs no need to call me fam, professor.
BPRP be like:
I donât even know these functions and I have a 90 something in calc 3
â@@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
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 didnât waste time not getting bored
I heard riemann zeta function so i knew that there would be Ï in the awnser. (Î(1/2) is also â(Ï) so that could've been a candidate aswell)
A circle be hiding somewhere in here.
đ§
Shall I email Mathologer?
ah yes the direct eta function, I'm still waiting for the mathematicians to release the indirect eta function
đ
đđđ
It's too op
Hahaha
"Dirish Lay"
Isn't it more like "Direesh Lay" ??
@@gamedepths4792 yep
I knew I couldn't do it đ
@@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 âDee-ree-klehâ. Thatâs 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, czcams.com/video/l7AErKEE9-4/video.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 ÎŁ 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.
It really did not disappoint. I would have never expected that "simple" outcome.
Glad you enjoyed 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
đ
This is like the final boss for integrals, you need your best equipment to conquer it.
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
Damnnn thatâs just beautiful, pi really shows up everywhere. Well done
Never seen this one before, this is crazy
Right?!
@@BriTheMathGuy yeah?
honestly i think the work (adventure) done in order to get to the answer is just as amazing as the answer itself
What has this crazy integral got to do with circles? đ€ If there is anything 3Blue1Brown taught me itâs 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). ÂŻ\_(ă)_/ÂŻ
my new way of doing integrals when the teacher asks me to do one on the blackboard: 5:15
That was nuts.
Right?
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. đđđđđ
The hardest integral Iâve ever done: 7 minutes
It takes me 7 minutes to find the integral of e^x đ
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..
â@@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
This was some heavy-duty content! Thumbs up
Thanks a lot!
This one is truly amazing broâ€ïžâ€ïž
Very glad you enjoyed it!
@@BriTheMathGuy â€ïž...
4:56 Dominated convergence theorem: âAm i a joke to you?â
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)
Thatâs such a beautiful problem!! Loved it
It really is! Glad you enjoyed it!
that escalated quickly. or rather it was already a bomb to begin with.
Your videos are really awesome! I've never enjoyed Maths until I started watching your videos keep up the good work đ
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.
â@@borsalinokizaru7382 that's crazy
So much information in so little time! Thank you sir
My pleasure!
Wow this is crazy man ! Literally took help of all three function to solve ! Great đ
Glad you liked it!
I changed my field of study 3 years ago. Im now a UX designer yet iâm still fan of calculus since it was the only talent i had and hated every thing else in engineering school.
1:26 "I''ll just call our integrand y"
"y who?"
"why wouldyouevendothat"
that's crazy good, thanks for the video
Thanks for watching!
Love your explanation.
So glad!
The editing on this video is superb!
Thank you so much!
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....
Cool! Definitely spreading this.
W o w.
Just Wow!
WOW!
That was so much fun!
That's insane!! What a great integral.
Just wow, thats so fucking awesome
My mind blew away. Need to make it once more.
That was awesome!
Glad you liked it!
I canât believe the CZcams algorithm took this long to recommend you to me. I love your channel! Thanks for the content.
O.M.G... this was awesome !!
Glad you thought so!!
Great one!
Thank you! Cheers!
well this was a wild ride...
Amazing video!!
Glad you liked it!
You're good. Keep it up !
Greatly enjoyed this one ...
Glad to hear it!
This is absolutely beautiful
Amazing!!
You are!
Iâm just waiting to see this as my final exam for Calc II
"Isn't it" hmmmm, a reference perhaps
2:50
Wow that is impressive!!!
Fun to watch, not so fun to solve - but an elegant solution is the reward!
Awesome bro
Thanks!
SOO much satisfying
I thought so too!
Crazy answer
Excellent
you mirrorly write ζ better than I normally write ζ
It involved the Basal Problem as a bonus too.
Magical
You look so proud to have solved this yourself
Thank you for using âlnâ to denote natural log
Awesome!
Thank you! Cheers!
This is why I love math
why do these ridiculous integrals have an answer related to pi lol
lol I wish I knew!
If there's e, you can find a way to end up at pi... Thank you Euler
because pi is a friend of all of calculus.
You are so good at integration
My mind is blown
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
yeah I trust you
It's not even 1 minute in and i'm having brain damage.
What an interesting problem and what an interesting solution.!!!!!
I like your videos !
Thanks! Very glad to hear it!
.......this is one of the most satisfying video for me.....
Great to hear!
2:31 " this is completely legal."
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 :) .
My brain exploded at the end
đ€Ż
When you said that the answer is beautiful, I knew it would have pi in it.
Letâs just appreciate how neat his mirror image handwriting is
it would be amazing if you do a video about lambert w function secondary branches
damn thatâs a lotta letters
Ok, u did it
Now take the same, but indefinite integral)
Best problĂšme ever
My brain stopped working
very nice, not as demandjng as i thought it would be
Glad you enjoyed it! Maybe I'll have to find something harder đ€
Give me a year or two and I'll be back when I understand
Pov: me as an 8th grader having a stroke seeing how difficult math would get in the future
me: wow i understand
my brain: I DO NOT UNDERSTAND ANYTHING IN THIS VIDEO
"Deer ish lay" -- Dirichlet
Sorry about that! đ
7 minutes of awesomeness.
Glad you thought so!
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.
Holy shit that was beautiful
Glad you enjoyed it!
The answer is pi^2 * (-infinity)
After watching this I only had one thought, how am I going to get past algebra?
You can do it!!
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!
That integral is pretty dope ngl
Glad you thought so! Thanks for watching!
how do you draw in the air is it glass? I'm confused. but a great video ( I understand the maths for sure)!
Itâs amazing when pi comes out of the nowhere
I graphed y=x^(xy) in desmos and the slope is so large it starts leaning left
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, ÏÂČ/12 is so beautiful answer đ
Jeez man, and then there I was thinking discrete math is hard