The Integral of your Dreams (or Nightmares)
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- Опубликовано: 20 фев 2021
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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.
#math #brithemathguy #integral
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At first I was amazed that he can do backwards writing so neatly. Then realised he just flipped the video
I was amazed but then you ruined the magic for me!!
What????
I was just going to comment the same thing.
that is pretty obvious...
@@manasaprakash7125 sarcasm dude 😅
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
truest thing i have heard
Next century physicist : hey guys, you will never believe what weird function I'm trying to integrate today
@@mathieuaurousseau100 - this century's pure mathematics is next century's applied mathematics, because of those meddling physicists.
😂So True!
@@BriTheMathGuy woah you saw my comment. Thanks bro you made my day 😊
Holy cow that’s the prettiest integral I have ever seen
I think so too!
👌
And he’s doing it backwards!
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 .
Never have I understood "Sufficiently advanced math is indistinguishable from magic" more than this very moment.
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.
@@GreenCaulerpa Yes thank you for explaining the joke. You get an internet cookie. Congratulations.
@@tnk4me4 yummy, thanks for that cookie!
@@GreenCaulerpa nice quote!
2:40 dude nice thank you for being aware that you can’t just interchange infinite sums and integrals willy nilly.
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].
@@HeinrichHartmann Series for e^x converges absolutely
@@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].
@@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
@@markusdemedeiros8513 thanks, I appreciate that summary.
That was NOT the result I was expecting form this. Absolutely beautiful
Glad you enjoyed it!
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.
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
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
You've fallen into my trap!!
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… 🤔
As an engineering student my first instinct was to use a euler's method of approximation cause "fuck that work" LOL
stupid approximateurs >:(
It’s ancient, but it works
You know you're an engineer when using π=3 does not seem like an approximation
@@chungus478, and a mathematics or physics student if it does.
Nice result, but now you should explain what is the value of the infinite sum 🛡️
It's maybe a bit late, but the value is round about 1.2912859970626636
@@johannes8144 Thank you! Did you compute that analyticaly or numericaly?
@@zebran4 It’s not possible to compute the value analytically as of this moment.
@@zebran4 By analytically you mean in terms of "non trivial" functions/expressions? If so it's very unlikely this can be expressed like that, just as a gut feeling
@@user_2793 Yes. By trivial expresions too.
we need more integrals like this, this is amazing
I really admire the way you explain, not in a hurry
Love your content! You can really feel your love for the math
Glad you enjoy it!
Wow, this was much better than i expected! Truly beautiful!
I have my term exams in few days and watching this is satisfying ❤️
Same here, good luck
@@tamazimuqeria6496 good luck
Best of luck all!!
How was it?
Well done! This is really amazzzing !
Amazing video!!
Thank you for sharing this beauty. Keep shining brother
You bet!
Wow, that was sum-thing else; thank you so much for sharing!
Glad you enjoyed it!
Friggin high school maths still giving me headache. Good job
Love the way you speak and write.
Thanks very much and thanks for watching!
Integrals Playlist!
ruclips.net/video/j0JN0q8FUtY/видео.html
It took 3 minutes for you to get to U substitution where I finally stopped questiining my sanity.
10/2 = {2,2,2,2,2} != 5
It's 1.29128
I am genuinely getting addicted to your videos !
Glad you like them!
Extraordinary! I didn't see it coming.
Could never work that out myself but it fun to look at.
This channel is amazing !!!!!!
blew my mind. Never seen summation and integrals after each other.
Pretty cool right?
@@BriTheMathGuy yeah but it also feels intimidating for someone who still has to pass his Calc 2.
You can do it though!
This was the cutest introduction of solution I have ever seen in addition to the handsomeness of the one who introduced it. 😅🤭 Bravo!
what a beauty!
Awesome!!! 😊
Omg the twist at the end is quite a shocker
An effective channel. Thank you
Glad you think so!
The screen inversion to get his writing right totally blown my mind to the point that I’m unable to focus on what he says.
Awesome vid! Good job!
Thanks for the visit!
You made it so simple :)
Glad you think so!
That is a very remarkable and beautiful result.
Even if you know all of these properties, there is so much knowledge that goes into applying them in ways that are helpful. Can't imagine figuring this out!
Good explanation!
Glad you think so!
Glass pane works really well. If you can dim the lights over your hand it will be much better.
That was beautiful - and scary!
Great video, cool result. Thanks for this.
Glad you liked it!
Wow, this is my kind of rollercoaster I enjoyed during lockdown, thanks math man
Glad to hear it!
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
Totally astounding
I find this so pretty. Almost like discrete sum (over all integers) of sinx/x = pi and integral (-inf to +inf) of sinx/x also equals pi. Amazing and yet baffling.
That answer is beautiful.
Math is so beautiful!!
7:08 moment of satisfaction
thank you
Welcome!
I'm here to comment just to make your video more popular
Thanks so much!
This might help me with a problem I’m working on
I've just finished with my Advanced Higher Mathematic course... just re-watching some of these videos for some good memories..
Great job!
0:17 Well, we don't _have_ to. The power rule gives xx^(x-1) = x^x, the exponential rule gives ln(x)x^x, so the total derivative is the sum: x^x + ln(x)x^x.
That works for x^x and x^(-x). But does this work for any derivative of f(x)^f(x)? Or only those cases?
@@qq3088 It generally works. It doesn't have to be exponentiation, and the functions don't need to be the same. It's a general property of differentiation that is used extensively. In other words, every derivative of a function with multiple instances of x can be realized as the sum of all "partial derivatives" with respect to each instance of x.
@@EpicMathTime l never knew this!
@@EpicMathTime "every derivative of a function with multiple instances of x can be realized as the sum of all partial derivatives with respect to each instance of x", damn that looks like a powerful statement. Do you know a proof for this?
@@dawnstudios7813 The simplest way to see this is to replace each instance of x with a separate variable (say x, y, etc), and take the total derivative with respect to t. Then, set x = y = ... = t. This collapses the total derivative to the special case of the single variable derivative.
This idea underpins differentiation very intimately. You're already doing it when you take any derivative, we just don't phrase it that way.
For example, let's take the derivative of sin(x)cos(x) using the statement you just quoted.
I'll treat the first instance of x as a constant, making sin(x) a "coefficient", so that 'partial derivative' is -sin(x)².
Now I'll treat the second instance of x as constant, and likewise, that 'partial derivative' is cos(x)².
Hence, the derivative is the sum of the "partials": cos(x)² - sin(x)².
Although I phrased it in this different way, what we did there is precisely the product rule. In other words, the product rule itself is a specific instance of doing the quoted statement.
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.
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.
@@captainhd9741 use desmos and input sum for sum and int for integral
@@jackweslycamacho8982 I prefer Wolfram but good idea!
@@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
The actually important explanation for interchanging sum and integral is brushed away like nothing. This took away the beauty of it.
I love the fact that a video about calculus was interrupted by an ad that talks about partials (dentiures).
😂
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!
I am a university student in Korea. I was always interested in math, and I happened to see your RUclips while I was looking for a related RUclips while preparing for a math test. I think there are a lot of fun and informative contents. I hope your RUclips will be better and I will continue to look for it often. Thank you!
do u guys learn calculus in high school in korea?
@@limagabriel7 Yes, I do learn, but for example, in the case of calculus that utilizes two or more variables, I learn properly in college.
Why are you named Apple Boss
damn that's fuckin sick dude!
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!
For an integral like that. You don’t get a numerical value
What I don’t understand is how mathematicians make such amazingly leaps such as the various substitutions to get to the answer.
Its a lot of trial and error, looking at past results and seeing if there are parallels, and a lot of luck :)
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?!?!?*
I like your funny words, magic man.
maybe more like a sophomore's nightmare to some i'd imagine
😂
I honestly think I'm more impressed by how good you are at writing backwards. LOL! Good video
He’s not writing backwards it’s just mirrored lol
I'm impressed by how well he can write mirrored then /jk
Nice solution. 👌👏
Thank you! Cheers!
I freakin love calculus. I thought this was gonna be really scary at first.
Just wow 🔥🔥🔥🔥
I tried thinking about this in a different way. I began by viewing the original (improper) integral as something I will call L (i.e., limiting sum for the improper integral). I take log(L) and then move the log operation on the inside of the integration. I doubt this obeys all the rules for logarithmic operations on (improper?) integrals. So now I am integrating the function -x log(x) dx on the same upper and lower bounds and still calling this L. The indefinite integral of this is computed to be (x^2)/4 - (1/2)(x^2) log x. Evaluating this at the limits gives 1/4 (the limit for the second term can be evaluated at the lower bound using rules for indeterminate forms and evaluates to a limiting value of 0, there from the right. Anyway, the upshot is that L = 1/4 which makes the original integral e^(1/4) or approximately 1.28, which is close to the result from the derivation in the video, but not identical. Why is this even close? I know something I've done must be wrong, probably because the integration must invoke the complex log function in some way, at least at the lower bound of integration.
It's ≈ 1.291291≈ 430/333
Newcomer here. How do you do that mirror writing thing?
uniform convergence is not sufficient to invert limit and integral, because the integration interval is not a segment (ln is not defined as 0)
*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*
Thats the best
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.
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.
There is, it equals sin(pi) / gamma(pi/2)
@@assasin1992m What is sine doing here? 🤔
@@assasin1992m makes me wonder if there is a complex extension for z^(-z) integral
Isn't sin(pi) 0?
@@ha14mu yes, but the limit toward pi in this expression converges to a non zero result
Amazing
You are!
Please upload videos on IMO problems too they are also very deep
Wow, just wow.
I was gonna discretize the domain and calculate the area by numerical methods.
can we try this for the indefinite integral of x^x dx?
Amazzing caculuses, guy!!!
Does the final infinite sum converge? Awesome integral btw!
Thanks! and yes it most certainly does! (around 1.29 or so)
@@BriTheMathGuy is there an exact identity for what it converges to, or did you just get this by approximation?
@@sophiophile there is no closed form for it sadly so all you can do is solve it numerically.
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.
@@tBagley43 almost all this kind of stuff has no closed form
Extremely interesting indeed.!!!!!
Glad you think so!
@@BriTheMathGuy I wonder if these functions need to be analytic or converge?
Beautiful 🤩
Thank you! Cheers!
What is the final value??
Would be interesting if the series which resulted from this integral converges to some value :) next challenge ? ;)
Is there a similar solution for the integral of the same function but fro 1 (or 0) to infinity. Should be convergent quite obviously, but how to calculate it?
Man! You love Gamma function so much 🤣🤣🤣🤣🤣
Yes I do!
How he interchanged the summation and integral signs at 2:30 please someone help me😢(btw i am class 11th student and jee aspirant)
excellent.
Thank you! Cheers!
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
What does the infinite sum evaluate to though? I looked it up on desmos and the integral evaluates to around 1.29. If we integrate from 0 to infinity, we get something eerily close to 2.
is it possible to approximate the sum as well? we know it converges.
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
Are you writing in mirror-image, or reflecting the entire video afterwards?
Can anyone explain what did he meant by uniform convergence.
Its intresting how he uses just SMALL PART of BOARD to explain such complex problems whereas for our teacher need two full boards
...oh dear i just got into this unit.
Excelente