An Integral Part of Happiness.

[@NPCooking69]

*Make sure to take a look over on **stemerch.com/** if you are interested in geeky STEM products! =D Get 8% off most of the products by using the code 420STEM at checkout! :)*

[@jkid1134]

Imagine writing this identity on a test and annotating "idk if this has a name but flammablemaths proved it"

[@PapaFlammy69]

xD

[@alexandersanchez9138]

Isn't this integral just a corollary of that super OP symmetry technique from the previous video? Edit: yeah -- ¡¡ retroactive spoiler warning !!

[@PapaFlammy69]

:DDD

[@mortadhaalaa5907]

2:20 Highest quality integral sign on this channel.

[@PapaFlammy69]

fucking fabulous

[@gabbarisback6052]

@@PapaFlammy69 o_O

[@neilgerace355]

0:00 i apples? Unreal!

[@PapaFlammy69]

It's complex :/

[@onthecover5042]

Fun fact: If you’re sad you’re not happy

[@PapaFlammy69]

ri🅱

[@integralboi2900]

Thank you Einstein, very cool!

[@mastershooter64]

Every 7 days in Africa, a week passes, together we can stop this.

[@kshitijsalunke2620]

hmmm... another one: If u die, you will not survive

[@BPEMEHA]

@@mastershooter64 I remember this line from Obama's inauguration speech, come on, give him some credit.

[@a_llama]

time for my daily dopamine and serotonin boost

[@PapaFlammy69]

[@sythlikesfrogs]

Kermit loves integration

[@PapaFlammy69]

hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

[@hoodedR]

@@PapaFlammy69 r/ihadastroke

[@PapaFlammy69]

xD

[@tecobel]

Can we just take a moment to admire how perfect the Integral symbol in 2:19 is?

[@PapaFlammy69]

:^)

[@tecobel]

@@PapaFlammy69

[@pierineri]

Summing the two fractions at 4:54 you may multiply numerator& denominator of the second one by e^x, and write it as e^x/(e^x+1) , whence it is clear that the sum is 1

[@dozzco2827]

Goddamn I love these videos, also this integral was brilliant, keep it up!

[@PapaFlammy69]

Thanks Dozzco! =)

[@a_llama]

this kind of looks like the even-odd integral technique you mentioned before! edit: nvm you brought it up lolol

[@PapaFlammy69]

:p

[@drumsNgames2112]

Papa flammy makes me appreciate my math degree a lot more. Keep the content coming.

[@Invalid571]

Hey papa flam. This question was on my latest Newtonian mechanics exam. I hope you can help me because I have no idea how to solve it. Here goes: Which of the following differential equations systems have periodic solutions: a) x' = y y' = 3x^2 - y - y^5 b) x' = y y' = x^2 + y^2 + 1 c) x' = y y' = (x^2 +1)y - x^5

[@PapaFlammy69]

I'll give it a shot! =)

[@Invalid571]

@@PapaFlammy69 Thanks! Less than three

[@user-un7gp4bl2l]

Geometry Dash Mega They are not linear equations

[@geometrydashmega238]

@@user-un7gp4bl2l I'm sorry you're totally right. I confused the dependent variables with the independent variable (I was thinking about laplace transform of t^n)

[@user-un7gp4bl2l]

Math & Coding I’m assuming periodic implies continuous. If this is the case, x’ and x’’ must pass through 0 periodically and when x’ is 0 x’’ has to take on both positive and negative values (otherwise the function would increase/decrease indefinitely). For b), x’’=x^2+x’^2+1, x’’>0 for all t (assuming real x lol), so that’s out of the question. A) is x’’=3x^2-x’-x’^5. At the points where x’=0, x’’=3x^2>=0, so x’’ cannot take on negative values there. So the only one that could admit such solutions is c), although I cannot prove that it does.

[@gtweak7]

The result at 5:42 is both hilarious and amusing, please keep those cool bois coming.

[@PapaFlammy69]

:D

[@blackhole3407]

0:21 #relatable

[@user-wu8yq1rb9t]

I have a question: What a about the odd functions ⁉️ For example (especially this one) if we have odd function in the numerator (like x^3 )❓(integration over zero to infinity)

[@sherlock2694]

Ein freund: Brüder, dieser equation is sehr schwierig... Ich: Was häst du dir gedacht? Ich brech' mir mein Genick? Bitte hören Sie mich das ich bin nicht eine noob.

[@Ferolii]

The Single Most Overpowered Integration Technique in Existence ^^

[@PapaFlammy69]

:p

[@francisrayquintana5119]

@@PapaFlammy69 You could call it SMOITE and it'll still work

[@MrRyanroberson1]

i'm watching this video immediately after the one on the identity related to e/(1+t^o) and it's so satisfying to just see it put into action right away.

[@taladon101]

Whole lotta crackhead energy lmaooooo

[@suga2420]

love the way you write your integral sign

[@PapaFlammy69]

Thx Suga :3

[@benschneider766]

An overkill method for such a sexy integral..NOICE

[@PapaFlammy69]

:3

[@vivekg8725]

Mad respect PAPA !!!

[@PapaFlammy69]

[@dysrhythmia]

this integration technique is so strong i had to manually write it into my book of integrals

[@robsbackyardastrophotograp8885]

In the Flammy's Method vid yesterday, you called the base in the denominator t, and didn't restrict it much. Today you presented it as an even. Is it only guaranteed for t=e(x) ?

[@PapaFlammy69]

Yup! I made a pinned comment stating it! =)

[@jamesmonteroso824]

int(x^n/(exp(x)-1)^k, x = 0 .. infinity) this can be solved by int(x^n*exp(x)/(exp(x)-1)^k, x = 0 .. infinity) and my technique of solving this is through integration by parts.....i do not know how to generalise it because the coefficients dont seem to have a pattern....Still they spew out rienmann zeta and gamma functions. Its interesting tho.

[@likestomeasurestuff3554]

Oddly satisfying. Overpowered integral generalization indeed...

[@PapaFlammy69]

:)

[@amryoussef3]

Good integral. If n is not an integer but is a positive real number, n is never equal -1/2. what happen if n=-1/2?

[@neorenhao2310]

Damn my PAPA done it again!

[@PapaFlammy69]

:)

[@MathIguess]

Yaaas new flammy video :D

[@PapaFlammy69]

Werner :000000000000000000000000000000000000000000000000000000

[@mohammadburhanmanzoor3806]

😱😱😱 symmetric boi!

[@PapaFlammy69]

ayyyyyyyyyyyyyyyyyyyyyyyyyy

[@danielescobar102]

I'm studying a double degree in math and physics Does that make me gae?

[@ricardoparada5375]

What would u even call the technique? Flammy’s technique of integration?

[@PapaFlammy69]

yeye

[@mastershooter64]

when are you gonna integrate the inverse gamma function or "ni" papa flammie pp?

[@PapaFlammy69]

Oh boi, that sht horrible m8 ngl ;_;

[@sohamchatterjee4288]

Out of 100 i would rate this only 5! ( if you know what i mean)

[@PapaFlammy69]

:p

[@gabbarisback6052]

5 factorial o_O

[@abhishekkp7121]

Another great integral. Next time do a video on solving the same integral in 10 ways.

[@PapaFlammy69]

:D

[@BPEMEHA]

the dots over the "i" on the right side of the t-short need to be situated vertically, not horizontally.

[@AlBoulley]

you got triggered, QED

[@andywright8803]

According to your clock, that was just one take!

[@cobollatin]

How is that theorem called? I am looking for the proof.

[@williamwilliam4944]

He has a video titled "the most overpowered integration technique" or something like that. The proof is his, since he discovered it

[@nikhilnagaria2672]

Came here from the even(x)/1+even(x)^odd(x) integration trick and can instantly tell it is int x^2n dx from 0 to t

[@PapaFlammy69]

nice! :D

[@uwiennr4686]

papi flam, maybe if u want, upload video about trading math with calculus ? 😂😂

[@atharvasharma3492]

Trieste comment 😁😁 I am in grade ten and I absolutely love your videos mannn Keep up the good work for making people love such a beautiful subject MATHEMATIC BYE

[@varunakmandokhot4439]

@Atharva Sharma forgive my curiosity, but how come you know integration even though you're in grade ten ? As far as I know, in India we learn basic integral calculus in grade twelve.

[@PapaFlammy69]

[@varunakmandokhot4439]

@@PapaFlammy69 yeah... I'm in grade ten too.

[@varunakmandokhot4439]

@@BerkayCeylan nice.

[@atharvasharma3492]

Varun A K Mandokhot I learned it through KZbin

[@amritbahl1223]

Uh- maze - ING

[@PapaFlammy69]

:3

[@mudkip_btw]

I love abstract apples mlem

[@mudkip_btw]

I wish apple analogies worked for the multiplicative identity :( if you have one apple, and you divide it in to 1 apple, idk

[@mudkip_btw]

That really is an overpowered technique :0 appreciate it.. need to go watch back that vid. Didn't have time :(

[@PapaFlammy69]

[@mudkip_btw]

@@PapaFlammy69 uwu

[@killerfc]

Oh 0 views and 3 likes.

[@PapaFlammy69]

nice.

[@mastershooter64]

@@PapaFlammy69 Illuminati confirmed!!! that's it shutdown the Doritos factories!

[@Jokster1]

Nice.

[@varunakmandokhot4439]

High five Papa Flammy! I have joined the Engineer hater's club too because of sinx=x, cosx=1, and e=π=3.

[@PapaFlammy69]

:'D

[@442277100]

but the euler's number is not an even number 🤦

[@nicholasthesilly]

not sure if troll

[@PapaFlammy69]

bruhv

[@442277100]

@@PapaFlammy69 omg i have been blessed by His mathematical presence 😳

[@pwootjuhs]

doctors tend to stay away from your videos I've heard

[@longphung4874]

Maths gang

[@mustafakalaycioglu9613]

Deja vu :)

[@PapaFlammy69]

:3

[@nathanmorgan3576]

Best

[@akshat9282]

who's yo daddy

[@PapaFlammy69]

ma mom

[@haradhandatta7048]

It will be short video , if you apply the formula: (integral from -a to. +a)f(x)dx=(integral from 0 to a)(f(x)+f(-x))dx. Please , think about this.

[@jjtt]

thank you random pajeet, very cool

[@ethannguyen2754]

I hate that shirt though

[@sahilbaori9052]

Aight, this apple is joke is getting annoying now ngl

[@PapaFlammy69]

xD