Flammable Maths By Tonelli’s theorem you can interchange the integral and the sum since your integrant is ≥0 on the interval you’re calculating the integral over ie. [0,1]
It's very intuitive because integrating a sum lets you split it into several integrals of each component of the sum. Then you can just sum up the integrals.
This is actually pretty amazing, because 1/(j^j) is essentially j^(-j), which is practically what we started with, x^(-x). So theoretically we converted a continuous sum (the integral) into a discrete sum of literally the same form! But instead of 0 to 1, we have 1 to infinity.
I wonder if this fact can be used to show that the set of real numbers is bigger than the set of natural numbers (instead of Cantor's diagonalization proof)
For people saying that this video is on my channel: I promise you that we came up with this idea independently a couple of weeks ago, it’s just a pure coincidence that our videos got uploaded almost on the same day!
When I’m faced with a hard Integral I usually just call the integrand f(x) and then I say let F(x) be the integral of f. I don’t know why you guys use these “techniques of integration”.
That actually does happen e.g. the special functions of analysis. Realize you can't find the antiderivative of say 1/ln(x)? Slap the name li(x) on it and call it a day
Papa Flammy, your way of explaining math problems is very unique and it goes straight to the head !! your attitude and way of explaining things is so cool and I never get stressed out.. wish I could get teachers like you in my graduation !! 😊😊😊
Why thank you!!!!!The mathematics channels are among my favorite youtube channels. Enormous amount of truth there!!! Amazing work and really well done and wishing everyone the best!!
I clearly have no words to appreciate you......i was like the type of person who hates integration the most....but the way u approached each and every questions bloomed up an interest in me to see more of ur videos....u really changed me a lot...i dont know whether you will see this comment or not...but im proud to say u r my frst subscriber related to MATHS...thumbs up bro..and GG LOVE FROM INDIA
I don't understand a lot about maths but it seems like substitution is a very important technique for proofs. Also, it's handy to know similar functions so you can look at similarities.
I barely understood what was going on and I LOVED IT! How did I just discover this channel? I love that the answer at the end was "like, 1.29something...?" Who cares about the answer when you have this beautiful process!
I’m just now coming back to this video 4 years later and now I fully understand it after taking calculus courses! I don’t even remember commenting on this video either.
Waaaaai. X^-x=1/x^x. We can call it f(x). So, integral of f(x) at 0 to 1= 1/1^1 +1/2^2... but, this is exaticly f(1) + f(2)... Edit: im sorry if us duficult to understand this, its because im Braziliam, im not so good at English
I just started learning calculus a few months ago, these videos are awesome because you do all the basic steps as well like at 1:13 , at my university the professors and teachers and even the university textbook never showed that step, it just showed 1:36 lol, great vids dude you make learning math seem just fine to a laymen.
lol reminds me that time when i asked on >implying what was the sum from n=1 to infinity of 1/x^x and someone said it probably didn't have a closed form but that it was equal to this integral.
You can do one more step! It is equivalent to Sum(x=1 to ∞, x^(-x)) The integrand of original integral is exactly same as the addend of the infinite summation!
I wonder now if it's possible to find a function whose integral is finite and is exactly equal to its infinite sum with the same boundaries (obviously one of the boundaries being infinity)
It's an interesting result that the integral of x^{-x} is equal to the sum of the very same thing with x taking discrete positive values. At least it's the first time I see a function with a property like that
So, you're telling me that the integral of a function is an infinite sum of infinitely small terms from that function. Sounds about right if you're looking to define what an integral is
I don't understand why moving from du to -du helps fix the problem of the natural log going to negative infinity. Now the infinity is positive but why does that help?
To y'all wondering why you can interchange sum and integral, and if I'm wrong please correct me: The sum(fn) series converges uniformly on an interval L, aka , with fn(x) = (-1)^n * x^k * ln(x)^k * 1/k! -> sum (sup[x€L] |fn(x)|) converges This condition is enough for the infinite boi and the integral to be switched.
Can you explain why the integral evaluated from 0 to infinity is so close to 2 but just a little less? Or why from 1 to infinity it's so close to 1/sqrt2? Maybe there is a nice explanation similar to the one with the Borwein integrals.
Another approach : Instead of using the Gamma function you could have noticed that u^k.exp(-ku+1) ist - k th derivative with respect to k now using the Leibniz rule you can bring rhe derivative out evaluate the integral and then get a nice formula for your infinity boi
Quite neat result. I think that (pedantically) one needs to make a convergence analysis when using Taylor's theorem to write the exponent as an infinite series as there is a potential problem near x=0. Nice video.
Two mistakes First, you just can't substitute ln0 as infinity in the integral limit Because it's undefined Second, you can't exchange integral limits without changing the sine of the integral
If i apply the riemann limit definition for the integral, i end up with very simmilar expression: \lim_{n\to\infty} \sum_{k=1}^n k^{-k} \frac{k}{n} ...
I think the answer might be -2. Try integrating it without introducing the sum notation. It's pretty amazing. Thanks for uploading this video, looking forward for more.
I wonder which functions, f, could replace x^-x in the first integral and final sum so that the integral from 0 to 1 of f(x) with respect to x equals the sum from k=1 to infinity of f(k).
3:36 Can you please elaborate on this(How you chose a substitution that converges). I'm new to this and would appreciate if you can give some sources to read more.
> assuming you can interchange integral and sum signs without proving it first
REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Flammable Maths By Tonelli’s theorem you can interchange the integral and the sum since your integrant is ≥0 on the interval you’re calculating the integral over ie. [0,1]
Philip Garmann
LOL
The main property required is that the integrand is integrable on every compact subinterval of the open half-line (0,\infty).
Hey we have the same first name :D
It's very intuitive because integrating a sum lets you split it into several integrals of each component of the sum. Then you can just sum up the integrals.
It‘s trivial😂
This is actually pretty amazing, because 1/(j^j) is essentially j^(-j), which is practically what we started with, x^(-x). So theoretically we converted a continuous sum (the integral) into a discrete sum of literally the same form! But instead of 0 to 1, we have 1 to infinity.
I wonder if this fact can be used to show that the set of real numbers is bigger than the set of natural numbers (instead of Cantor's diagonalization proof)
TheGamer583
Why don't you try doing a Riemann Sum computation of this integral?
=1.29129
@@PavelSTL , a look at "the Hilbert hotel" ( &/v at "the hyperwebster" ) will give you a negative answer ... !!!
Wonder what other functions have \int_0^1 f(x) dx becoming \sum_{j=1}^\infty f(j) :) Great video Flammable Maths!
Sorry sir, I couldn't hear you over the sound of my engineering approximations
Mr. Skeltal hahahaha
Trigonometry for AS and A level th-cam.com/video/-WzZRx4vVxI/w-d-xo.html
Watch and share and subscribe our channel
Drinking game: take a shot anytime he says "boi" or "boiz"
Oliver Hees j dond ghe vhallenve alr2adg it went fins i fhink
you ll solve riemann conjecture but only drunk people will understand you ll forget the solution when sober
this is your destiny
Crack open a cold one with the boiz
"Infinity boi" is now my new favorite phrase
Who would win? The proof of Fermat's Last Theorem or
O N E I N F I N I T I B O I
I love when the Gamma Function pops up in these crazy integrals.
Infinity bois are the best bois
wait wtf memes and maths?
This boi is the only English speaker boy that uses the pacman :V
Kudos from Mexico xd
It's like oil and water, or rather a poly protic compound and a non-polar compound
Chemistry bois will know this
For people saying that this video is on my channel: I promise you that we came up with this idea independently a couple of weeks ago, it’s just a pure coincidence that our videos got uploaded almost on the same day!
Dr. Peyam's Show haha I was thinking "I already saw this one". You should have made it in German ;p
Can you do a video on nonlinear PDEs ?
i dont believe it
Oh boy here we go again, it's the Leibniz-Newton controversy again.
calculate for me the probability of that happening
This was fun! You're a natural teacher - I look forward to seeing what you will upload in the future!
hello simooooon :D i have no idea how i found your comment in this video, it was randomly recommended to me lol
I sense fanfiction probabilities...🙈
Top 10 anime crossovers
Ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh myyyyy
I agree dr. Clark!
commuting an infinite sum and integral without hesitation *im shook*
My real analysis professor would've had a heart attack seeing this.
and +Elfarouk Harb: This could be proved using the Fubini's theorem.
@@hansenchen1 he should have atleast written it
@@hansenchen1 Elaborate please
@@chihebsaidi2054 Der Beweis ist dem Studenten als Hausaufgabe aufgegeben
Now that's the 'recommended for you' I was hoping for
When I’m faced with a hard Integral I usually just call the integrand f(x) and then I say let F(x) be the integral of f.
I don’t know why you guys use these “techniques of integration”.
And here we have a physicist
It is a standard method of solving...what you said... I.e we take F(x) is the integral and then simplify... We have a whole chapter of 45 pages on it
That actually does happen e.g. the special functions of analysis. Realize you can't find the antiderivative of say 1/ln(x)? Slap the name li(x) on it and call it a day
Exam: Find the solution to this equation
Me: Def: x a real number such that it satisfies this equation.
x is the solution.
@@appa609 well that's kinda what the root is in a sense. sqrt(n) is the positive number x such that x^2 = n
This was fun to watch. When he did substitutions so it fit perfectly into the gamma function template and things started canceling out I was blown
This is how good I want to be one day.
Hello, im from future! Have you achieved the goal?
I laughed.
I cried.
You, sir, are an artist
BadJumpCuts, it's called modern art.
Imprecision is art?
this video is making me relive uncomfortable traumatic experiences from calc 2. so glad I'm in the cozy arms of simple calc 3
the BOI
Papa Flammy, your way of explaining math problems is very unique and it goes straight to the head !! your attitude and way of explaining things is so cool and I never get stressed out.. wish I could get teachers like you in my graduation !! 😊😊😊
Why thank you!!!!!The mathematics channels are among my favorite youtube channels. Enormous amount of truth there!!! Amazing work and really well done and wishing everyone the best!!
Me after finishing AP Calc: Damn I can finally watch videos on calculus and know what’s happening.
Nvm
Hahahahahahaha u thought lmaooooo
Putnam memes for Sophomore dreams
I clearly have no words to appreciate you......i was like the type of person who hates integration the most....but the way u approached each and every questions bloomed up an interest in me to see more of ur videos....u really changed me a lot...i dont know whether you will see this comment or not...but im proud to say u r my frst subscriber related to MATHS...thumbs up bro..and GG
LOVE FROM INDIA
I love how int(x^-x dx,x=0,x=1) = sum(j^-j,j=1,inf). Beautiful symmetry there
I have no words to describe how beautiful the final result is. Also, this was one of your very well made videos :)
Beautiful :)
Dr. Peyam's Show hey!
Three of my favorite mathematicians all in one spot! Yay!
46 and pi yay!!!!!!
Yee
you lost me at infinity boi
I don't understand a lot about maths but it seems like substitution is a very important technique for proofs. Also, it's handy to know similar functions so you can look at similarities.
Dankest maths channel I have ever seen 🐸👌🏻🎩
:3
Its 1:30 AM I'm high and watching this. Dudeeee
I barely understood what was going on and I LOVED IT! How did I just discover this channel? I love that the answer at the end was "like, 1.29something...?" Who cares about the answer when you have this beautiful process!
please keep doing these beautiful integrals
So beautiful solution with your rythmical good tempo as well as german like intonation and pronunciation. I love it.
ah so you're an English major
I’m just now coming back to this video 4 years later and now I fully understand it after taking calculus courses! I don’t even remember commenting on this video either.
Waaaaai. X^-x=1/x^x. We can call it f(x). So, integral of f(x) at 0 to 1= 1/1^1 +1/2^2... but, this is exaticly f(1) + f(2)...
Edit: im sorry if us duficult to understand this, its because im Braziliam, im not so good at English
I think it's surprising that the integral from 0 to 1 of x^(-x) is exactly the sum from 1 to infinity of x^(-x)
I just started learning calculus a few months ago, these videos are awesome because you do all the basic steps as well like at 1:13 , at my university the professors and teachers and even the university textbook never showed that step, it just showed 1:36 lol, great vids dude you make learning math seem just fine to a laymen.
lol reminds me that time when i asked on >implying what was the sum from n=1 to infinity of 1/x^x and someone said it probably didn't have a closed form but that it was equal to this integral.
We have the same way of reasoning and learning as far as I have seen. You don't make many analogies, I love your videos, keep going mate.
It is very interesting the way your final result looks almost the exact same as your original problem, just in a discrete form
There is some interesting semmetry between the initial problem and the solution
You make me smile a little bit every time you say 'infinity boi' :) subscribed!
Now this is my favorite video of your channel
I like the way he pulled the -1^k /k! in from the cold !!
I find it interesting that the integral from 0 to 1 of x^-x dx ends up being the sum from 1 to infinity of j^-j
Nice job man, I really liked it and you!
thx Robert :)
I started panicking when you wrote dx = -e^u du, but I calmed down when you subtly rewrote it as -e^-u du
xd
xD
You can do one more step!
It is equivalent to Sum(x=1 to ∞, x^(-x))
The integrand of original integral is exactly same as the addend of the infinite summation!
Yes, that's a mesmerizing result!
integral from 0 to 1 of 1/x^x dx = sum from 1 to infinity 1/k^k
cool
...I just can't see an intuitive geometric argument to explain why "0 to 1" became "1 to infinity".
I wonder now if it's possible to find a function whose integral is finite and is exactly equal to its infinite sum with the same boundaries (obviously one of the boundaries being infinity)
It's pretty neat that the integral from 0 to 1 of x^(-x) is just equal to the sum from 0 to infinity of x^(-x).
Really interesting integral, thank you! The nice thing is that we integrate x^(-x) and get sum of k^(-k), very symmetric thing :)
The more I see those integration puzzles the more I love numerical methods.
So the integral of this function from 0 to 1 is just the infinite sum of the same function from 0 to infinity? There is something more to dig there.
This function have optimization task answer is e^-e by away it second example where e=2,718 have minimum value property.
It's an interesting result that the integral of x^{-x} is equal to the sum of the very same thing with x taking discrete positive values. At least it's the first time I see a function with a property like that
Literally liked the video before I even started watching it
First video to reach 100k+ views. proud of you :)
It's me, I used my mom phone to watch and commented, now you already passed 100k subscribers, very proud of you papa flammy, time flies~
Can you do a video explaining the interchange of the sum/integral and show some examples where it doesn't work? Or do a video on uniform convergence?
So, you're telling me that the integral of a function is an infinite sum of infinitely small terms from that function. Sounds about right if you're looking to define what an integral is
The integral that was massaged into a gamma function was also just the Laplace transform of u^k evaluated at k+1
i n f i n i t y b o i
Find someone who looks at you like our boi looks at the integral at 6:06
what the fuck even is math
How have I only just found this channel? It's awesome!
Hey, can I ask what your 'day job' is?
I don't understand why moving from du to -du helps fix the problem of the natural log going to negative infinity. Now the infinity is positive but why does that help?
Couldn't you just leave it a positive du and make your lower bound for the next integral negative? That seems more natural to me
i love the way you teach x) if my lecturer spoke like this i would be laughing none stop
These videos make me so happy!
That (-1) getting canceled out was one of the most satisfying things I've seen in some time lately
To y'all wondering why you can interchange sum and integral, and if I'm wrong please correct me:
The sum(fn) series converges uniformly on an interval L, aka , with fn(x) = (-1)^n * x^k * ln(x)^k * 1/k!
-> sum (sup[x€L] |fn(x)|) converges
This condition is enough for the infinite boi and the integral to be switched.
Can you explain why the integral evaluated from 0 to infinity is so close to 2 but just a little less? Or why from 1 to infinity it's so close to 1/sqrt2? Maybe there is a nice explanation similar to the one with the Borwein integrals.
Flammable Maths thanks for your quick response although i still don't believe it's a coincidence.
Keep up the good work👍🏽
i think without doing this any one can do it simply by taking y=(whole integration )
and then take log on both side and then apply product rule
Generalize inegral(a,b,f(x)) = integral (a,b, g(g-1(f(x))) g is a function with some properties
Infinity boi
Memes and maths. Subscribed.
I saw your videos thinking I was subscribed lol, definitely your charisma makes me concentrate completely on the video haha
The integral from 0 to 1 of a function is equal to the summation of that function from 1 to infinity, that function being x^-x
Hmm. What other functions have the same property? We should name them "flammable functions", or something.
Damn I miss calc. Saw this video on recommendation, very easy to follow and nicely explained.
prob bois saw the reciprocal of the scaling factor for a gamma density right away
Another approach : Instead of using the Gamma function you could have noticed that u^k.exp(-ku+1) ist - k th derivative with respect to k now using the Leibniz rule you can bring rhe derivative out evaluate the integral and then get a nice formula for your infinity boi
Bro can u show us how to integrate x^( 1/x )
GammaFirion, and from zero to infinity
Hint: Let u=1/x
Is this some meme I’m not getting?
Lol, i thought you used j for your jmaginary unit
I am happy , I was able to do this question in another way.
Love this content and love your videos, thanks math boi
just stumbled into this (after 5 yrs). But isn't this "just" the reciprocal value of the Bernouilli integral?
Quite neat result. I think that (pedantically) one needs to make a convergence analysis when using Taylor's theorem to write the exponent as an infinite series as there is a potential problem near x=0.
Nice video.
Two mistakes
First, you just can't substitute ln0 as infinity in the integral limit Because it's undefined
Second, you can't exchange integral limits without changing the sine of the integral
If i apply the riemann limit definition for the integral, i end up with very simmilar expression: \lim_{n\to\infty} \sum_{k=1}^n k^{-k} \frac{k}{n} ...
Another great example. I need to watch how to solve an integral from here everyday. That will make my day!
I think the answer might be -2.
Try integrating it without introducing the sum notation. It's pretty amazing.
Thanks for uploading this video, looking forward for more.
I showed this work to my lil sister and just laughted and said "what.s his favourite pokemon?" :))))
There is a mistake ... between 6:50-7:00 ... e^(-uk).e^(-u)=e^(-uk-u)=e^[(-u)(k+1)]
But in solution k-1 is written...
Sorry...i dont realize that your plus is like minus 😁
no problem:)
nice development. Not very difficult, but you have to make the right magical substitutions. Congratulations.
Why am i watching this again...I just saw it two days ago, and now this is the third time. Oh well. Hi Papa!
Glad to see you back then Ranjan! xD
I wonder which functions, f, could replace x^-x in the first integral and final sum so that the integral from 0 to 1 of f(x) with respect to x equals the sum from k=1 to infinity of f(k).
6:00 sooo when you switched the limits of the integral, you just took the -1 of the e^-u and you didn't touch the signs of the other terms?
I have no clue how to do calculus and I usually understand about 10% of what's going on but I love this guy's videos
:)
I just scrolled down to see the bottom blackboard. I am not a smart man.
3:36 Can you please elaborate on this(How you chose a substitution that converges). I'm new to this and would appreciate if you can give some sources to read more.
Honestly the title is why I'm here. Subbed.