A very interesting integral with aesthetically pleasing solution development

Today's integral looks absolutely gorgeous and yeilds a very cool final result in terms of our favorite transcendentals.
My complex analysis lectures:
• Complex Analysis Lectures
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[@maths_505]

If you like the videos, feel like you're learning something and would like to support my efforts: www.patreon.com/Maths505 You can follow me on Instagram for write ups that come in handy for my videos: instagram.com/maths.505?igshid=MzRlODBiNWFlZA==

[@noobymaster6980]

I love when he said ”it’s integral time” and integraled all over the integral. Truly one of the integral moments of 2024.

[@edcify8241]

You're the only one who didn't get a heart

[@CalculusIsFun1]

@@edcify8241you just had to jinx it.

[@lyonkang7402]

I thought we were done with this

[@CM63_France]

Hi, "ok, cool" : 1:06 , 5:27 , 6:37 , 7:37 , 8:12 , "terribly sorry about that" : 2:20 .

[@boranxiii]

damn this video was so beautiful that i lost my edging streak.I love math.

[@maths_505]

Yup KZbin is definitely recommending my videos to the right audience.

[@BenDover69831]

bro

[@NavyaMenon25]

guys i think he likes likes math

[@josephlorizzo8997]

WOW, what a cute ODE, i loved how you used the feyman trick there, i couldn't wait for you to mention the gaussian integral for that -u² on the e ahahahaha

[@ieee12345]

It feels like it'd be a ton easier writing cos(x) = Re(exp(ix)), packing the exponentials, completing the square and then invoking some mild holomorphic property to make the imaginary shift in the integration variable. After that it's just gaussian integrating. You could even do that at the step before, changing the sin into a complex exp. Still, watching Feynman's trick at work is always nice, keep it up.

[@lakshay3745]

Or you could just use the complex definition of cosx and then complete the square in both Integrals and create two erf functions which solves the integral in like 2-3 lines

[@aryaghahremani9304]

can you explain the holomorphic shift? what i get when completing the square is Re(exp(-(x^2-i/2)), how do you turn this back into a normal guassian without invoking complex integrals

[@ieee12345]

@@aryaghahremani9304you should get something like exp(-(x-k*i)^2) inside the integral, for some k I can't tell right now. Then you'd want to do the change of variables u = x - k*i, to get back to the normal gaussian integral, but to do so you must note that the 2 "line pieces at real infinity" integrate to 0 and that the integrand is holomorphic. You can also just do the change, replace the limits and hope it works, but to justify it you need this very mild complex analysis, which is what I was referring to.

[@djconnel]

@@lakshay3745 I agree: I felt as if the use of Fenyman's trick was strained. My first reaction was to express the cosine in complex exponentials, as you suggest. But maybe because I never learned it in school so I find it non-intuitive.

[@Sugarman96]

And at that point it's just the Fourier transform of the Gaussian evaluated at 1

[@askandpushpaltiwary8537]

you should try the book of almost impossible integrals. Its a joy to solve those!

[@utuberaj60]

Hi Kamal I've been watching your amazing 'integral' videos, and the more I watch, the more I love them, and I see you using that amazing Feynman's trick that's really cool. Indeed you make my 'integral moment' today at sunrise here in India

[@aryaghahremani9304]

ODE is smart, i love how feynman's technique is becoming an actual method of solving integrals rather than just a cool trick these days i solved the cosx e^(-x^2) integral by letting cosx=Re(e^ix) though which is pretty cool as well i guess

[@zealot4325]

Thank you!

[@Jalina69]

Counting how many times bro said "cool".

[@nickyhekster2974]

Very cool.

[@holyshit922]

Integration by parts Change of variable u=sqrt(-ln(x)) Series expansion for cosine Change order of integration and summation Change of variable v = u^2 , to get Γ function (Γ function can be also expanded)

[@giuseppemalaguti435]

Io ho fatto così...t=√(-lnx)..risulta e^(-t^2)cost integrata da 0--->inf...poi feyman I(a)=..cosat...risulta,in sintesi I=√π/2e^(-1/4)

[@Ivan-mp6ff]

Why not complement it with a graph? In most math calculations, a graph speaks a thousand words and can be very intuitive. Thank you all the same.

[@sasukeuchiha1320]

Ok cool!

[@federicoformaro]

one of the few times i could've solved it by myself

[@lucasbeaumont5762]

people say im weird cause to me this is fun, i think their werid for not understanding that this was fun

[@MrWael1970]

Very impressive integral. Thanks for featured solution.

[@cameronspalding9792]

@ 4:49 for this part I would have used the complex expression for cos(u) and then used contour integration to get the rest of the answer

[@HighlyEntropicMind]

I liked how we never needed to go back to the original

[@acelm8437]

I liked how the e^(-u^2) term kept absorbing the u's

[@lolcollol212]

Free okay cool buttons: 1:05 5:27 6:36 7:37 8:11

[@BadrBoutara]

hi please why did you plug in ln c rather than c

[@davidblauyoutube]

Ah, but is the final result irrational?

[@dukenukem9770]

Both the solution and the technique are beautiful.

[@illumexhisoka6181]

I have read about norms the other day (barely understood anything) But I got wondering how to find a general formula for the integral from 0 to 1 of The nth root of (1-x^n)

[@GiornoYoshikage]

Substitute `x^n = u` and notice that you get a kind of beta-function

[@trelosyiaellinika]

Really cool! 😊

[@leonmancaj3690]

Great thank you Sir

[@KarthikDasari-g5q]

i thought you sove it by complex nos

[@maths_505]

Keepin it real this time

[@toufikakkak8459]

Hi bro, can you do the intégrale from 0 to 1 for "ln(1-ln x)"

[@maths_505]

Aight

[@sciencelover-c2j]

When you get in the next step, please put some explanation between steps. Because we need to get the clearance solution. we are not all proficient .thank you

[@maths_505]

So you weren't listening to the audio???

[@sciencelover-c2j]

@maths_505 Absolutely, because mathematics is a language in itself, so l asked you, we need more steps before you get in another step,

[@maths_505]

@@sciencelover-c2j to waste more time??

[@sciencelover-c2j]

@maths_505 Why do you think that 20 minutes is not enough ??

[@maths_505]

@user-zg8ny5tp4g I just think that rambling on about basic algebra is gonna be extremely boring. The level of math here is something that my target audience is sufficiently familiar with.

[@YassFuentes]

Beautiful one!

[@igorbatkovich3856]

Без Фейнмана не обошлось))

[@SamLindskougMBGY]

First time i accually manage to solve one of your monster integrals, lets go

[@maths_505]

I'm proud of ya bro

[@yoav613]

Nice😊

[@albert3344]

Ooooook coooool!!!

[@edmundwoolliams1240]

Beautiful aesthetically pleasing result, use of differentiation under the integral sign which led to a lovely first order ODE 😍 Exactly what I needed on my Friday morning. Could you find and do a beasty integral that evaluates to Digamma(G/sqrt(phi))?

[@maths_505]

Wow that is some request! I'll try my best.

[@edmundwoolliams1240]

@@maths_505 I know that's a very tough request. 😂 I don't expect you to get very far with it; as far as I'm aware crafting a (non-trivial) integral to yield a specific result is some near-impossible guesswork. I suppose that's why it's extra-special when beautiful results DO come out 😊

[@2739D]

mane wtf😭😭