The Single Most Overpowered Integration Technique in Existence.
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- Опубликовано: 21 сен 2020
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Odd Even Decomposition: • Decomposing functions ...
Even function over symmetric interval: • Even integrand over a ...
Odd function over symmetric interval: • Integration technique:...
Method Applied: • A pretty awesome integ...
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Today we use odd and even decompositions to derive an extremely powerful real integral identity. Enjoy this amazing boi!
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*The answer is around 6 or 7... let us just approximate it to 1*
I died
Especially considering the answer is actually around 30
~1 great meme, ngl
Lmao same
as an engineer, your weakness disgusts me
If it's around you can round
As a physicist myself I appreciate your talent for wild rounding. Thats something I can get behind.
:D
Engineers too
@@TheLucidDreamer12 you forgot to approximate your comment
Astrophysicist: order of magnitudes only.
Can you suggest to me how to be good at Physics? I want to be a Physicist in the future. Please suggest me.
Haha wolfram alpha can't find a closed form of the final integral
ladies and gentlemen: we gottem
:DDD
Thats not actually true. I typed it in and wolfram got the right answer
@@NightwindArcher yea 'cause it numerically integrated i.e. cheated
Ladies and gentlemen *We gott'em*
@@duncanw9901 Seems like this is implemented in Wolfram Mathematica. It finds (exactly) correct answers to integrals of this form
My first instinct when I see an integral like this is to check if the function is odd.
same :)
My first instinct is to never encounter it.
If not, just 👑
My first instinct is to inhale chlorine gas
I’m 100% going to troll my teacher with this one but then get a zero because he didn’t understand it
rip
0:15 Phlegmmable Maths
*r i p*
F
9:18
"And now we're going to create ourselves an absolute abomination of an integral and we are just going to rip its ass wide open"
I DIED MATE
:D
Isn't t(x) restrained to being an even function (not any function), because, if it's not, t(x)^o(x) * t(-x)^(- o(x)) = 1 is not valid?
Oh, probably! Thanks for the constructive comment!
Yes, it is necessary that t(x) = t(-x) in order for the result derived to be true.
t doesn't have to be continuous though as long as that property holds 😈
@@PapaFlammy69 i think t should be restricted to be non-negative on the interval, so you can apply that t^o × t^(-o) = 1 for all x :)
the example indeed does work out though
So now we have math formulas to come up with clickbaity thumbnails ?
Now that's meta.
Congratulations on getting references on Wikipedia bro, really looking forward to your proof of e=1
I can't believe the official Wikipedia article on the lists of integrals links to this video lmao
So wait...it happened 2 months ago?
Whoa!
"official" Wikipedia article. Do you even know how the site works?
so tragic someone removed it. fucking vergins
@@sofielundsskolan You clearly don't. :)
Last time I was this early the Pythagoreans were killing Hippasus.
ezzzzzz xD
Me before watching the video: *1 + 1 = 2*
Me after watching the video: *so we can integrate everything there is, as long as we have Papa's brilliant method*
:D
"Good moOOOrning fello mathe - UHGHOUGGHHAAAHH- video"
I died of laughter😂
:D
wow that was quick
indeed :v
8:02 Ah yes O - O = 0
The three applications of o, O, and 0 fucked with me the entire time
xD
For the annotation at 9:25, I think you mean t can be any even function of x. If t isn't even, you don't get the same simplification to 1 when combining fractions.
Yup, already noticed!! Thanks for the comment though Steve!
The first time I saw f(x) written as a sum of an even and odd function was in Paul Nahin's book "Inside Interesting Integrals," and this was just as spectacular as when I first read it. The idea is so clear in hindsight, that it begs the question: Who thinks of this stuff? Wish it was me lol. Great stuff.
:)
“What the fuck Papa” earned my subscription. Well done sir
The intro is top tier
Flammy this is a great video. Everything delivered in point!
bruh they gotta nerf this next patch.
and fix all the bugs with 0 and such, but this is OP as heck.
:D
That's okay. We were talking in the comments about how t(x) needs to be even for this work. A fair nerf, though the integral is still OP AF
@@angelmendez-rivera351 t doesn't need to be even though? We never assumed t to be even. It can be any function.
Rohit Tanwar Papa Flammy never made any explicit assumptions in the video about t(x), but the algebraic manipulations he used to simplify the integral only work if t(x) = t(-x). In particular, he inadvertently simplified t(x)^o(x)·t(-x)^[-o(x)] to 1, but this simplification is only valid if t(x) = t(-x).
The integration meta will never be the same
I love the concept of this video.
I learned how I should think about integrals ( or in general about Math).
Great Papa Flammy, Great.
I Learned and enjoyed
Thank you Papa ♥️
lets start the video with some corona yes
x'D
I love the enthusiasm! Thank you for a nice presentation of an even more nice technique :-)
Your enthusiasm and sheer happines(like at 10:33), makes watching your videos me enthusiastic and happy.
Glad you liked the video Benjamin! :3
Thanks for the video! Always love to add another technique to my repertoire.
Any chance you could go over Maclaurin Integration? It's new (published this year) and works for every integral as long as there's not a discontinuity between 0 and 2. Maybe you make a sequel to this video like "The Most Overpowered Integration Technique Pt. 2" or something.
Anyway, I really enjoy your vids, thank you for the work you contribute to the community! ♥️
Here before the physicists
ez
That's true
Hey i am a physicist
@@physicsboy1234 youre still late
@@minh9545 i got first or second comment
7:13 rotating head of James Grime on fire 😂😂
:D
RUclips : an Interesting report you have made there citizen
How didn't I notice that?
How come i haven't derived such a good shortcut. Great work man. Thanks for this.
This is lovely. (I'm delighted to say, incidentally, that Mathematica handles the integral in the thumbnail!)
:)
WHo eLSe iS cOMinG frOm WiKIPediA?
I don’t see it on my Wikipedia so I am checking it here
I think it got removed
@@wanderingpalace NOOOO it really got removed
who came back here and watched the video again after this getting referenced in the wikipedia article for the list of integrals
The most useful thing I ever found bro...
My adrenaline while watching this went 📈📈📈📈📈📈📈
LOVE YOUR VIDS!!
Great integral!
thx Zerg :3
ruclips.net/video/4tlNSfM53h0/видео.html
Very nice--This is excellent stuff! A truly OP integration technique. It's amazing how powerful symmetry can be in the right circumstances.
(Yes, yes, I've already noticed that t needs to be even as well so you guys don't have to tell me.)
:) Glad you liked it Alexander! :D
Blackpen showed the exact same technique a year ago :
ruclips.net/video/gK8Wf9ZkZW4/видео.html
0:15 that is the most wonderful intro to a video I have witnessed
Best video in a while, always love the channel
Thx Ian! :)
Well that's *_one_* way to start a video!
Came here from Wikepedia, stayed for the whole thing because I remembered who you were seeing your face and hearing your voice as opposed to seeing you channel name!
Kudos ç
✌
Hello from wikipedia
Me about to start hw: 9:22 - 9:24
Me starting hw: 10:14
LMAO
That meme in the beginning gives you the motivation to listen to the entire video.
That clock is some next level stuff🔥
:D
Thanks for this video. Extremely interesting
YO PAPA this is CHAD mathematics right there
i love the enthusiasm here so much
2:16 "Well we are going to have an even function in the numerator and an odd function in the power"
I think we all know where this is going
:D
This guy is the pewdiepie in the math community
*Pewdieπ
First vid I saw of yours, good shit
no way dude nice technique, wow always some nice stuff in integration techniques man
That's a nice one! Thanks for sharing!
=)
i know, it has been noted a few times already, but t(x) needs to be an even function. the problem however comes when t(x)
Surprisingly it always holds, even if the even function got poles! :D
this type of integral actually appeared in the 2023 stanford math tournemant lol and i was able to use this trick to solve it! thanks :D
Can't express myself enough, so just thank you so much for the joy of doing maths.
:))
Really cool use of even and odd functions
thx! :)
I absolutely love this, brilliance, nothing short!
Can't wait to forget about this method during the January exams :P
Hands down the best papa Flammy intro
This is amazing, I’m in awe !
Glad you enjoyed the video René!
man , i love your channel
:3
I just rewind and rewatch the intro over and over XD
As an engineer, I didn't notice anything wrong when he said "well e is basically equal to 2"
"Pi is absolutely clickbait, it must work out" -A wise flammy
:D
Who comes from wikipedia?
Where's do you get these integrals... although great though:)
Dunno, I come up with many of these out of boredom :D
Is there any website which I can refer to...
i think there are exceptions around poles, so you have to determine where along your interval the denominator is zero. this occurs for pi^sin(x) = -1, so we have arcsin((1+2k)ipi/ln(pi)). this is not in the path of your integral so you're good to go.
Hey Papa :) Stemerch looks dope
Thx Ranjan! =)
I had to turn in my lab protocol today and spent the whole day writing this bullshit and drawing graphs.
I needed this video, thank you
t can’t be anything, we assumed it was even during the derivation
Best intro so far! :D
:D
I have calculus exam later. This is very helpful! Well done Sir/Prof/Ma bui
gotta add THIS IS VERY FCKING HELPFUL KEK
:D
You should mention that t can be zero since integral is over [-a,a], but use symmetry to do [0,a] (as you did) and then just do (0,a] instead. Not an issue, but I'm surprised you didn't mention it! :) good vid
Just kidding, for some reason I thought t was the integration variable xD disregard
This result is amazing! What inspired its original derivation, if I can ask?
Brute force tbh ^^'
You fired it really 👑👑
I threw up a little bit when he used "o" as a function. I know it stands for odd but, its just so gross
For t to be the same base with -x and x t must be an even function from the proof you gave - but it is nice
yup!!
What would you do if the denominator wasn't 1+t^o but rather some other number? like 2+t^o? is there a way too solve that?
This might be my favorite video ever. I was laughing my ass off!!!!
=D
This is too OP for my mind to comprehend
This is my new favourite Video!
I'm pretty sure that if t is not a constant it has to be an even function because it has to remain the same when you plug in -x.
yup!
coolest trick I learnt today, I'll use it if I get a test on it
:)
The decomposition of a function into even and odd is exactly what the complex definition of sine and cosine do they split the exponential into an even part and an odd part can every nonsymmetrical function like rational radicals be split into even and odd?
This is amazing. Did you invent this method?
What do you think, e^(pi)(i) -1 =0 or ceil(e) - floor(pi) = 0?
Ես ինչ բարդ ձևով ապացուցիր, ավելի հեշտ ձև կա ախպերս:
Watched the title: WTF!!
Watched the video to the end: 😁👍
That's so amazing!!!! So creative :D Where di you learn it from?
Came up with it myself =)
@@PapaFlammy69 How??? D:
can you integrate sin(sin(x) - x) ftom 0 to 2*pi?? From its graph looks like the answer is zero but why? thanks.
Nice question. You have f(x)=sin(sin(x)-x). Now shift this curve to the left by pi units. You now get f(x+pi), call this new function g(x).
Applying this transformation: g(x)=f(x+pi)= sin(sin(x+pi)-(x+pi)) = sin(-sin(x)-(x+pi)) = sin(-sin(x)-x-pi) = sin((-sin(x)-x)-pi) = -sin(pi-(-sin(x)-x)) = -sin(sin(x)+x)
The final result is that g(x)= -sin(sin(x)+x). *Note* that I used symmetry properties: sin(-x)=-sin(x), and sin(pi-x) = sin(x), and sin(pi+x)=-sin(x).
Since g(x) is just f(x) shifted left by pi, wouldn't you agree that the integral you stated is equivalent to the integral of g(x) now from -pi to pi? (we are just shifting f(x) to the left by pi and now integrating the exact same thing).
Well then, we can also show that g(x) is an odd function. g(-x) = -sin(sin(-x)-x) = -sin(-sin(x)-x) = sin(sin(x)+x) = -g(x). Hence it is odd.
Well since g(x) is odd, then the integral from -pi to pi is just 0, as the area is just reflected in the y axis. Hence your integral is equivalently 0.
If anything is unclear just ask.
Another way of saying this is that if you take the y axis to be at x=pi, then your function is odd. (And we show this by shifting the function left by Pi, and showing that it is odd about the y axis)
funnily enough if you ignore the /5 part, the result would have been more accurate. e^5 /5 ~ 30, which means a guess of 2^5 was spot-on. e / 5^.2 ~ 1.97, which is quite close to 2
How can variable t be any function of x? It only works if it’s an even function. If it’s odd or a mixture you don’t get that nice cancellation.
They say when you round off things you create another reality. That makes that thing variant.
~TVA 2021
Nice. Hope I can get to use this technique once.
hopefully! :)
Why is there another integral on preview? When we have cos(x)^(sin(x)) instead of π^sin(x) we cant integrate in +/- e bounds because of cosine negativity out of range of (-pi/2;pi/2).
Wow you are just like my old maths teacher, but with less cursing. :)
watch my older videos :p
Papa, for your comment at 9:24, t can't be ANY function of x, as then you're going to have t(-x), which isn't going to cancel out so nicely in the end.
Yup, it needs to be even! =)
If papa Flammy were a fisherman he would damn well be a fishing magnate
:D
6:59 you could have instead simplified the second fraction by multiplying by t^o which will give you the same denominator, no mess :)
Does this type of Integral show up on year 1 calculas
Can you make a playlist of all the integration techniques you've come up with
I could try to do so! Check the description of the video back in a few hours! =)
Actually this thing is insane. But i have one question: what if the function E or O or both were not to be integrable between minus a and a, we cannot use the linearity of the integral and thus cannot say that integral of E plus O equals integral of E plus integral of O and eventually equals to integral of E. For instance, let a>0 and let the function E equals -1/x^2 and O equals 1/x. E is even and O is odd and neither of them are integrable on the line segment [-a;a]. Thus, writing integral of 1/x-1/x^2 equals integral of 1/x minus integral of 1/x^2 has no sense at all because the both integrals are not defined. So my question is the following: does this works with any kind of even and odd function, or is there any restriction to it?
Cool learned new thanks mannn btw that cough was nice😂😂😂
:D
Flammable Maths 😄😄😄😄 btw huge fan!!!
Thx Atharva!
Flammable Maths 😁😁😊
that was so fire that my house burned down...