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"This is completely legal" is exactly what a criminal would say.

Bored student: “Prof? I wanna see an integral with all my favourite special functions!!” Professor: “Say no more, fam!”

A circle be hiding somewhere in here.

ah yes the direct eta function, I'm still waiting for the mathematicians to release the indirect eta function

Spoiler alert: The answer will not disappoint you. I recommend watching it to the very end.

Many of you may be wondering, "how does Brian know that the tower converges on the interval [0, 1]?" Let me help answer this question. In this other video that Brian made a while ago, kzfaq.info/get/bejne/op1xeNV8qqudXWg.html, I wrote in the comments how the power tower, with base x, converges when x lies in the interval [1/e^e, e^(1/e)]. In this video, though, the base of the power tower is x^x, rather than x. The minimum value of x^x is (1/e)^(1/e), which occurs at x = 1/e. Because 1/e < 1, (1/e)^e = 1/e^e < 1/e < (1/e)^(1/e). Additionally, 1 < e^(1/e). So x^x takes on the values of the interval [(1/e)^(1/e), 1] when x lies on the interval [0, 1], and since [0, 1] is contained in the interval [1/e^e, e^(1/e)], the integrand actually converges to the explicit expression for the given interval of integration. The power series is also guaranteed to converge for the same reason, since the convergence of the power series for W is interdependent on the convergence of the power tower. So in fact, every manipulation Brian used to evaluate the integral is valid, and the integral is indeed well-defined, because the integrand is well-defined on the interval of integration.

It really did not disappoint. I would have never expected that "simple" outcome.

"Dirish Lay"

This is like the final boss for integrals, you need your best equipment to conquer it.

What are *your* favorite/crazy integrals?!

'Im challenging you to make it through the video' Me who doesn't know any calculus and watches as if its in another language: Easy

It took me longer than I care to admit, but... I managed to derive the integral at 5:10 by making the substitution t = - (n+1) ln(x). This make the integrand (-1/(n+1))^(n+1) t^n e^(-t) dt, and the bounds of integration are from infinity to zero. That substitution didn't show up out of thin air, by the way. I got there by first trying t = ln(x) (which gave the exponent the wrong sign), then t = x ln(x) (which required the Lambert W function just to rewrite the integrand), and then t = - ln(x). With that last one, I wound up with the integrand (-1) t^n e^(- t (n+1)) dt, so I first tried t = - ln(x)/(n+1), and then realized I needed t = - (n+1) ln(x). I added that rather long-winded explanation to illustrate that sometimes, you need to try several things when solving a problem.

Damnnn that’s just beautiful, pi really shows up everywhere. Well done

honestly i think the work (adventure) done in order to get to the answer is just as amazing as the answer itself

What has this crazy integral got to do with circles? 🤔 If there is anything 3Blue1Brown taught me it’s that whenever there is pi in your answer there is a crazy link to circles (although it would be sad if many cases have links to circles which go unnoticed due to cancellation like pi/pi).

Never seen this one before, this is crazy

Been watching your calculus videos for a long time, you explain things very well. I learned a lot of calculus, just by watching your content. I hope that you continue to make more of them, I write down all the examples you do. 👍👍👍👍👍

my new way of doing integrals when the teacher asks me to do one on the blackboard: 5:15

I changed my field of study 3 years ago. Im now a UX designer yet i’m still fan of calculus since it was the only talent i had and hated every thing else in engineering school.

that escalated quickly. or rather it was already a bomb to begin with.

That was nuts.

That’s such a beautiful problem!! Loved it

Wow this is crazy man ! Literally took help of all three function to solve ! Great 👍

The hardest integral I’ve ever done: 7 minutes It takes me 7 minutes to find the integral of e^x 😭

So much information in so little time! Thank you sir

Your videos are really awesome! I've never enjoyed Maths until I started watching your videos keep up the good work 👏

This was some heavy-duty content! Thumbs up

I can’t believe the KZfaq algorithm took this long to recommend you to me. I love your channel! Thanks for the content.

4:56 Dominated convergence theorem: “Am i a joke to you?”

The editing on this video is superb!

This one is truly amazing bro❤️❤️

I’m just waiting to see this as my final exam for Calc II

Love your explanation.

W o w. Just Wow! WOW! That was so much fun!

My mind blew away. Need to make it once more.

That's insane!! What a great integral.

well this was a wild ride...

that's crazy good, thanks for the video

1:26 "I''ll just call our integrand y" "y who?" "why wouldyouevendothat"

Fun to watch, not so fun to solve - but an elegant solution is the reward!

"Isn't it" hmmmm, a reference perhaps 2:50

Cool! Definitely spreading this.

you mirrorly write ζ better than I normally write ζ

I actually visited your site and i wanted to know under which course i will be able to learn about advanced integration like you have shown in the video and under which category it comes under? because i am really curious to learn about new functions. btw love your content and ya i made it through the video :) .

O.M.G... this was awesome !!

Just wow, thats so fucking awesome

It involved the Basal Problem as a bonus too.

You look so proud to have solved this yourself

This is absolutely beautiful

That was awesome!

You're good. Keep it up !

it would be amazing if you do a video about lambert w function secondary branches

Thank you for using “ln” to denote natural log

Amazing video!!

Can you please provide some stuff on Legendre and Bessel function. Thank you

Great one!

Greatly enjoyed this one ...

Do you have a Gaussian integral type representation for x^^x? (here ^^ is the tetration symbol, x should be real and positive) If yes, what is the derivative of x^^x orthe integral of x^^x in a positive interval? (e.g. from 0 to a, a>0)

Hello sir , being a student writing JEE ADVANCED , i would like to hear your advice for cracking the math part .

SOO much satisfying

how do you draw in the air is it glass? I'm confused. but a great video ( I understand the maths for sure)!

Crazy answer

What happens if you change the limits to e^-e and e^1/e? :)

1:45 hold on for a moment, if we do that we first have to know if the limit converges, how do we know that over x in {0,1} that it converges? 4:50 what about the other branches for Lambert W? I can barely find anything on the internet discussing them nor their series representations

Amazing!!

When you said that the answer is beautiful, I knew it would have pi in it.

Wow that is impressive!!!

I kept clicking on the videos you recommend at the end and now im stuck in a loop, pls help

How much time should I spend on solving algebraic problems to write new stuff by wave of the hand?

Does anyone know what happens when you change the binds of integration to something like 0 to 0.3?

I graphed y=x^(xy) in desmos and the slope is so large it starts leaning left

This is why I love math

I am not very aware of those functions cuz i am school student, but i have seen them and understand a bit about them, so yeah i got through the video. I think i now need to figure out what is lambert function and etc :) My teacher taught us about gamma function but i think i need to figure out more :D Thnx Also, π²/12 is so beautiful answer 😂

Magical

It's not even 1 minute in and i'm having brain damage.

Let’s just appreciate how neat his mirror image handwriting is

What an interesting problem and what an interesting solution.!!!!!

My brain exploded at the end

I am not really that good at math but I just had a feeling the integral probably had something to do with pi, and I was amazed when I saw the final answer!

How did you plug it in the lambert Function at the end I am trying to understand for the past 30 minutes

This is a beautiful solution! I can't say I completely understood it, but the result is so surprising. By the way, I wonder what's the average age of your viewers, so if someone here in the comments has any idea, I'll be glad to hear.

My brain stopped working

My mind is blown

Excellent

Is there any conditiom that can change the order of integral and infinite sum?

Give me a year or two and I'll be back when I understand

After watching this I only had one thought, how am I going to get past algebra?

why do these ridiculous integrals have an answer related to pi lol

3:56 alternate: w(w(x)e^w(x))= w(x) on the other side, when you take w(x), w(x) = w(x). Solved!

but can you actually switch integral with infinite sum?

You are so good at integration

Best problème ever

.......this is one of the most satisfying video for me.....

Where did the -1+n exponent come from in the x ln(x) substitution? 4:52

Wait... you got a constant from an integral? Or was there more stuff to do?

damn that’s a lotta letters

Ok, u did it Now take the same, but indefinite integral)

You should've shown what it looks like on the graph

Awesome!