One Sick Integral!

Wanna prepare yourself for more Integarahls or just feel like trying out new & interesting problems? Why not try out Brilliant then? =D brilliant.org/FlammableMaths
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Integral 1/(1+x^4): • What a crazy approach!...
Leibniz Rule for Integration: • The Leibniz rule for i...
Gamma Function: • DERIVING THE GAMMA FUN...
Euler's Reflection Formula: • Euler's Reflection For...
Today we take a look at the integral from 0 to infinity of 1/1+x^n , which by using Feynmans Integration Trick, unfolds nicely into a Gamma function fest! :D Enjoy :3
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[@PapaFlammy69]

*Thanks for watching my dear children :v Hope you enjoyed this one :)* If you feel like supporting the channel, make sure to check out some of today's relevant links, aight? Wanna prepare yourself for more Integarahls or just feel like trying out new & interesting problems? Why not try out Brilliant then? =D brilliant.org/FlammableMaths Infimum Merch :0 papaflammy.creator-spring.com/listing/infimum? Integral 1/(1+x^4): kzbin.info/www/bejne/eXq1eIhjj8hsnpY Leibniz Rule for Integration: kzbin.info/www/bejne/rZzLYoxth5amhdk Gamma Function: kzbin.info/www/bejne/ZmKaYYdrqt6Zq68 Euler's Reflection Formula: kzbin.info/www/bejne/eWK3fnilZpd3f7M

[@anshulrai677]

Nice

[@sohammanna9090]

Ok !!!

[@anshulrai677]

Second comment first(baba flammy)

[@MathbyGaabriel]

Yass. Papa. I don"t speak English but te amo. 🔥🖤⚔️

[@Jared7873]

Why is this so complicated? 😳

[@epicmorphism2240]

That‘s why I subscribed, to see how integrals get fckin destroyed

[@PapaFlammy69]

[@Devilextremeful]

Sounds hot lol

[@koenth2359]

We don't need no water let the integarals burn.

[@AndrewDotsonvideos]

This changes everything.

[@akselai]

get a load of THIS late boi

[@veralgupta8182]

Actual title : sick integral gets cured

[@30indrayudhdas28]

Atlast, some integral stuffs after long time!!!!!!😁😁😁😁

[@goodplacetostart9099]

His english is so great that no one can imitate the way he say π as pa

[@leonhardeuler7647]

Finally, after a long time, an Integarahl! :)

[@HershO.]

0:05 lol the intro memes are always creative it's almost like this person put these numbers in a fkin calculator and got infinite😂😂😂

[@CreativeMathProblems]

There's a much quicker way to do this. Use the Beta Function, after making a tan substitution of x^n = tan²(x). Try it, you'll have to use Euler's reflection formula but it'll then give you a very nice quick way to do this

[@PapaFlammy69]

that method is boring, that's why I didn't show it

[@duggydo]

Papa's camera focus trolling him still. 😎

[@PapaFlammy69]

ye 3:

[@mastershooter64]

1:10 Feynmann is the one who popularised it, so I think the math bois and the physics bois should come to a compromise and call it Leinbniz-Feynman integration method

[@PapaFlammy69]

no.

[@joda7697]

@@PapaFlammy69 yes

[@Godakuri]

Did Feynman popularize it because of quantum mechanics?? I’m self teaching quantum mechanics and differentiation under the integral is never referred to as Feynman’s technique. But I am aware of Feynmans contribution to QM

[@angelmendez-rivera351]

@@Godakuri Differentiation under the integral is not the same thing as Feynman's technique. Differentiation under the integral is utilized by Feynman's technique, along with other ideas, but the differentiation is not the technique itself.

[@angelmendez-rivera351]

The theorem that allows for differentiation under the integral is due to Leibniz, but the parametrization method by which one uses the aforementioned theorem in order to create a differential equation with initial conditions uniquely solved by the desired integral is by Feynman. They refer to different things, mathematically, so it makes sense that they would have different names.

[@kaipar9743]

You explain wayy better than my maf prof. Ty pappa flammy

[@PapaFlammy69]

[@LetoTheGodEmperor]

This answer is so beautiful I’m going to cry

[@Diamond-ji2gv]

This guy is the kind of guy who says: if I buy more than 1 board imma use all boards

[@ExtremeAgent]

Papa Flammy: "One Sick Integral!" Me: Y E S

[@theanist3908]

Your videos are really great!!!

[@Ryunichi464]

You can also just substitute x^n to some sort of tan²(t), from there we can easily get rid of the denominator. And from thus on, manipulate it a bit so that we can use Beta Function.

[@KillianDefaoite]

Very creative method. This integral is a real beast to tackle with the residue theorem, I was pleasantly surprised to see that it has a more direct solution, even if it involves a clever parameterization.

[@fragileomniscience7647]

Was sagt ein Algebraiker, wenn er Privatsphäre braucht? "Das ist mein Eigenraum!"

[@PapaFlammy69]

:DDD

[@shekhar9750]

Nice to see wonderful process

[@whatby101]

Love the content, just learned several of these concepts recently. Shocked this integral has such a nice general formula.

[@PapaFlammy69]

:)

[@whatby101]

@@PapaFlammy69 Not exactly expecting you to see this, but I want you to know I have watching your content since the end of highschool. I am now taking complex analysis as my first graduate level math course, and will be taking two more graduate courses next semester. You are one of the math youtubers who really gave me the edge in courses. Thanks again for the content, it always keeps me excited about math, even when I was stuck on a problem for hours earlier in the day haha.

[@laxminarayanbhandari855]

This is a very easy integral. I had solved this a month or two ago, on Quora. But he did it a bit differently than how I did. I just substituted t=x^n and used beta. Then exactly what he did.

[@xerostat8961]

my last brain cells are ded after watching your board at 7:40

[@beardydave926]

The slick board changes, adequate use of "is hence nothing other than" and engineering approximations convinced me that this is correct and sick.

[@gg.3812]

I actually like to do this kinda integrals with contour integration. It is so much more intuitive to guess the functions to introduce just by thinking about the kind of asymptotics needed and the solution often comes so much quicklier

[@vatsall]

Dear Papa, I'm in medical school now but your videos are still as excellent as when I started watching you in high school. Much love, keep up the good stuff!

[@PapaFlammy69]

[@Rundas69420]

Waited too long for stuff like this :3

[@alessandro.calzavara]

10:49 Sheldon Cooper moment

[@mudkip_btw]

Hell yeah this makes me very happy :D good morning flammu-desu

[@tszhanglau5747]

Nice integral, very cool

[@BRORIGIN]

Damn integral videos taking me back

[@peterdriscoll4070]

Feynman would be proud. Amazing result.

[@cyberdash]

That parameter choice is so clever

[@brendanlawlor2214]

Fantastic integral without complex analysis / residues kept it simple that's a great teacher 😃👍🇦🇺🦘🦘🦘🐨

[@mathematicsmi]

If we used beta function it is very simple and few steps.. also another method- use Ramanujan Master theorem..

[@Evan-ne5bu]

Yooo this integral has some drip ngl 💯

[@brunoparracho8102]

This is beautiful

[@tszchunlau223]

You're back after a week :D

[@allanwalker105]

Bro you are explained very best I can understand

[@shivaudaiyar2556]

This is an easy one we are waiting for a flammable one hope will see it soon😁😁😁

[@krumpy8259]

These special conditions to check for to make a change in the original expression, could elaborate on them in a series of videos, since you already made a bunch of rather sophisticated examples of integrals where things must be given in order to make the changes. I know they are annoying, but when studying math these conditions are always more important than the actual integral and it helps to deal with those kind of proofs. Thanks.

[@juandiegoparales9379]

so beautiful, thanks

[@acecreation4770]

Aayy I was just doing some trigonometry and it's nice to see some integrals here😀😂😂

[@MathElite]

Super interesting video

[@mathematicsmi]

Nice evaluation... I will also make an video for this question by using another methods..

[@clementboutaric3952]

This is truly gorgeous.

[@divyaraval1909]

Awesome 😌

[@josewui414]

This is such a sexy answer dude. Easily verifiable by n = 2. Nice job papa.

[@koenth2359]

Ingenious genious!

[@matteovissani1071]

Missing the old integral times.

[@mastershooter64]

papa flammy plz do a video on differential forms and wedge products

[@jonathannissim2774]

thanks man great integral. love from tel aviv university

[@PapaFlammy69]

[@blackdeath39muffin45]

Lolz got hyped when noticed we can use reflection formula

[@davidblauyoutube]

I know you said you didn't want to use complex analysis, but the contours of integration have such a pleasing geometric symmetry and the residue theorem makes the problem quite simple.

[@riadsouissi]

I think it is more impressive to use this method (Feynman) as well as a contour method to prove the Euler reflexion formula instead. Nice Integrahal video!

[@jcfgykjtdk]

truly sick

[@infinity_1223]

Wonderful👍

[@steliostoulis1875]

Missed you papa

[@gamingstars8956]

Integral boiiiii🔥🔥🔥

[@theflaggeddragon9472]

Could you do some videos on Euler's proof that sum_{primes} 1/p diverges and its connection to the analytic class number formula? Gives some really neat formulas for pi in terms of Dirichlet series!

[@Grak70]

That was a SPICY integral 💕 🌶

[@jensbauer1141]

Really nice solution :D But can you solve the seemingly more complex Integral from 0 to Infinity of 1/(x^n +x^1/n). The solution is nearly similiar to this one, but I don't know how to make the Rechenweg... Would be interesting to see :D Thanks for your nice content!

[@affablegiraffable]

Something about your accent makes this so fun to listen to

[@ricardoparada5375]

Yo that final answer looks pretty cool

[@neilgerace355]

0:07 On my HP-15C, the highest number is 69. 69, dudes!

[@GAPIntoTheGame]

What I love about this is solution is that if n is very big then pi/n is very smol and using the fundamental theorem of engineering we say that sin(pi/n)=pi/n, then the solution becomes: I=pi/(n*sin(pi/n))=pi/(n*pi/n) which equals 1

[@heylofellas]

I mean sinx/x as x tends to 0 is also 1 lol

[@heylofellas]

But aight, message received

[@Afdch]

5:50 "...under the condition nothing is equal to zero, We're going to assume t to be strictly positive" 12:03 "well, if we just say let t be equal to 0" :thinking:

[@WhattheHectogon]

Ausgezeichnet!

[@bezirksbesamer9309]

pls destroy more integarals in the future papa

[@sushantsingh8700]

E=MC2 M=E/C2 M=W/C2. (replaced E=energy in form of W=work done) M=VQ/C2 M=VIT/C2 M=I2RT/C2 M=V2/R*T/C2 Hence, M=V2T/RC2. Where, E=energy, C=speed of light, W=work done, V=potential difference, Q=charge, I=electric current, R= resistance, T=time. I apologize for sending physics stuffs in your class. But please check this derivation is it correct or not.

[@123atulpandey]

I haven't studied this. I m in grade 11 and we aren't taught about it but I still found it pretty much fun😁

[@Godakuri]

They don’t teach this in elementary calculus. At least they didn’t for me

[@arnavagarwal2914]

I don't think they teach this in schools

[@Grak70]

You are here prepping to devastate your college calculus courses. Keep it up.

[@nicholasheilig3694]

Amaizing

[@dharmatejareddy9640]

Papa, make a video on integhraal for x^3/(e^x -1) with out contour. Danke schön Papa flammy

[@zplays4347]

UwU flammy integrals finally come back :'D

[@trypophobia7295]

What am I doing watching this while I’m still learning how to differentiate x

[@erikawimmer7908]

Sorry that I'm late Papa. I had an appointment pls dont beat me.

[@PapaFlammy69]

[@shahinjahanlu2199]

Great

[@antoinecantin1780]

Oddly satisfying lol

[@dave_xc]

Yo can you come up for a formula for me? I have a steel sphere with a diameter of 12.7 mm and I drop it from a meter and I want to know how strong is the wind velocity by the distance from the place perpendicular to the ground of the sphere in my hand and the place where the ball dropped

[@doraleemonhedt7079]

what is the intro music? sounds very pog

[@gtweak7]

Please come Integarahl's back.

[@gigagrzybiarz]

Noice integharal

[@alperakyuz9702]

Papa flammy can you pls recommend me a good book on pde's?

[@lucapalese475]

nice

[@drsonaligupta75]

Papa flammy!!!!

[@PapaFlammy69]

Alphabet!!!

[@jonathangrube1183]

I don't understand how we can first assume t to be strictly positive during the substitution and later set t=0 when integrating with respect to t? Do we need to let t approach zero from above and is this just left out here or is there something else going on?

[@PapaFlammy69]

we let t go to 0+ in the limit

[@mckg2992]

Pls make more videos about Fresh Toadwalker, lol

[@ShaneCasserly]

I love you

[@danielkirilov8065]

Hello Papa Flammy, I follow you since you were under 10k followers :^3. I have a question. In one of your videos before you were evaluating an integral which you took from a book. As much as I remember the book was with yellow cover and was only about hard integrals with their solutions. You showed the book in the video and also gave us the link in the description. I can't remember the name of the book. Can you try and find out what was it? Also you had a video with recommended math books but I can't find it :^(

[@mathematicsmi]

I don’t know the book you mentioned, but read this book , “ almost impossible integrals, sums and series” by Cornel loan, one of my fb friend, and “insight interesting integrals” by Paul j Nahin, also “ The harmonic series and logarithmic integrals “ by Ali Shadar who is also one of my friend.. if you need I can help you.. just email me.. I will send PDFs..

[@danielkirilov8065]

@@mathematicsmi Thx a lot!

[@danielkirilov8065]

@@mathematicsmi The first book you said is the exact one I was searching for so long time. Thank you.

[@danielkirilov8065]

@@mathematicsmi I found them online thank you.

[@einsteingonzalez4336]

0:04 Here's an interesting inequality. (\frac{x+1}{2})^{x} > x! for x ≥ 2. Thus, we find that 85.5^171 > 170! So upon calculating it, it's ~10^330. Thus, 171! < 10^330. Therefore, the answer isn't ∞.

[@adamharoon6021]

Yes, 0:26 Pi=7.

[@argyrispap4999]

Can you explain fermats last theorem proof ?

[@emiltaraldsen8988]

You said that you let I(t)’s domain be strictly positive so that dividing by t wouldn’t be an issue, but When you integrated I’(t) you used t=0 as the lower bound, isn’t this problematic? I might be wrong, just wondering. Love your content by the way

[@dhritimannath3282]

Love from India bro ❤

[@Mephisto707]

For what range of n does this result hold?

[@LamantinoElettronico]

0:05 Isn't it just the largest number whose factorial fits in a double precision floating point number, and thus the highest factorial you can calculate with a 64 bit computer?

[@MathbyGaabriel]

❤️

[@korigamik]

Does't it seem to break if we use n = 1. I mean how would we connect cosec and log together? Someone, help me!

[@nicholasthesilly]

Yes and no. The original integral diverges at n=1, so this still gives the correct answer. It just happens to be infinity.

[@zh84]

Very nice! Here's an easy puzzle. I have an infinite number of cubes with sides 1, 1/2, 1/3, 1/4... If I line them all up they stretch to infinity (the harmonic series). What is their total surface area?

[@alessandro.calzavara]

10 (I'm an engineering student)

[@zh84]

@@alessandro.calzavara Correct to 1 decimal place.

[@mathematicsmi]

@@zh84 glad to see you here... Today I watched this video.. after watching this I motivate to solve this integral in different ways.. suddenly my mind goes to Ramanujan Master Theorem..and it will another elegant solution...

[@oni8337]

Do some abstract algebra lol

[@David-km2ie]

Abel ruffini theorem

[@Jared7873]

Do you want abstract algebra to explode your brain? 🤯

[@oni8337]

@@Jared7873 I want to learn some tho

[@vaibhavcm7503]

Hello sir, I tried solving the alternating harmonic series. But I'm stuck. I took it to be the summation of (-1)^n/(n+1). And I write (-1)^n as e^n(ln-1) and I wrote ln(-1) as iπ and then I used Euler's identity and wrote is as the summation of (cos (nπ)+isin(nπ))/n+1 and then on breaking the summation the the summation which contains sin(nπ) becomes zero. And we get summation of (cos(nπ))/n+1. But I don't know what to do next. I'm stuck on this from days. Could you please help me sir.... please.....