the hardest integral from the BMT integration bee

The Fall 2023 Berkeley Math Tournament will be held on the UC Berkeley campus on Saturday, Nov. 4th and I will be there! For more info and registration, check out berkeley.mt/events/bmt-2023/.
I solved this monster improper integral during the Extreme Calculus Calculus (another 100 integrals). Full video • 100 integrals (almost ...
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[@blackpenredpen]

The Fall 2023 Berkeley Math Tournament will be held on the UC Berkeley campus on Saturday, Nov. 4th and I will be there! For more info and registration, check out berkeley.mt/events/bmt-2023/.

[@BradleyG01]

This is why I love this channel. Takes an integral that, quite frankly, almost makes me wanna cry, and turns it into something where I just need to brush up on infinite series to understand.

[@roastedchicken5197]

I’m laughing when he said “I know this has the N but that’s ln” his facial expression and the way he delivered is killing me😂.

[@wallace7815]

Same

[@blackpenredpen]

Lol😂

[@kiss6917]

HAHAA SAMEE

[@parikshitkulkarni3551]

😂

[@chovie]

Ya, and what makes it extra funny to me is that these types of jokes he makes might have come from IRL mistakes he saw at some point :D

[@-.SkyArt.-]

Looks so complex yet it’s so simple! Thanks for the tutorial, you never disappoint. 😊

[@blackpenredpen]

Thanks!

[@-.SkyArt.-]

@@blackpenredpen No problem! Also I wanted to point out that I’ve been watching you for a while now and never expected you to see any of my comments, let alone reply to one! Thanks so much for replying, it made my day ❤️

[@maitland1007]

That was fun. Thanks. I thought it was interesting that the integral itself wasn't that hard, but rather the simplification of the series. Have you ever thought about getting refillable pens? They work great.

[@Hamzakhan-km5cc]

One of most unique integrals I’ve seen in a while

[@bowenjudd1028]

I wouldn't have been able to do this on my own, but man is this one satisfying

[@blackpenredpen]

I did this integral after 100 integrals lol full video kzbin.info/www/bejne/oILdYpqHZ5mCfsU

[@ericgolightly8450]

1,000,000 SUBS

[@en2-joserivera896]

Hi.

[@EPMTUNES]

I love your videos. Over the years you’ve always been so helpful, so direct, and so charismatic. Even if I don’t understand what’s happening it’s still entertaining. Thank you

[@blackpenredpen]

Thank you!

[@joyis9638]

That was fun! I have not done calculus classes since 1980 as a mathematics major but I could follow this! Great work and explanations!

[@shrayanpramanik8985]

Feels great to be able to solve it while being in high school. Thanks bprp. You started my year old obsession with integrals🙇‍♂️.

[@freedomofmusic2112]

EXCELLENT VIDEO, thank you for this. I struggled so much with infinite series, it's nice to see a fairly practical application of it to solve an incredibly difficult problem

[@Starlight-sc4bp]

I passed the hardest topics in Calculus thanks to you. Love how detailed your videos are.

[@mathsrelated]

🤣🤣7:49.... He has done his work & there's no reward except this mad behavior.

[@jedhcurtis1148]

This is just beautiful!!! Love your work!!

[@puh8825]

It's so simplistic and beautiful, what an awesome integral

[@puh8825]

On a side note, we would be screwed if we didn't have Euler's symbol

[@misinagy983]

THAT WAS AWESOME! thanks bprp :)

[@kingarth0r]

Very fun integral, adding it to the list!

[@qillerdaemon9331]

7:50 "And that's a good place to stop."

[@gewinner3455]

Amazing video! So much fun to watch!

[@augustus5169]

I am currently learning series and sequences in high school and this was so cool to see!

[@kpax9284]

you have series and sequences in Highschool curriculum? Isn’t it college level?

[@tinyeragon7132]

@@kpax9284 its taught in AP bc calc, taught in high school

[@lincolnwithrow8951]

@@kpax9284 Pre-calc junior year most likely.

[@mohammadaminbanaei9237]

And even better, e - 1/e is W^-1(1) + W^-1(-1) W(x) is the lambert W function

[@aadhavan7127]

2sinh(1) is even better imo

[@quantummathematics777]

​@@aadhavan7127 😂😂😂

[@user-vt4bz2vl6j]

FISH

[@nirajabcd]

Not doing maths anymore but it is a sheer joy to watch you doing all these interesting problems

[@AT-zr9tv]

Same here. His joy is relatable, for 20-year old me.

[@marius4363]

this is a beautiful one , like after seeing how its done it dosen't seem hard but it would probably take me days before actually seeing the Maclaurin Series for e^x with the ln x being there, props to the guys that found this beauty

[@ravikumargautam72]

This is incredible solution 👍

[@eitancahlon]

It is probabbly the most beautiful integral I've ever seen.

[@mohamedsamir9527]

Perfect as always 👏👏

[@binaryhaze-fm6lk]

Amazing mathmatical thought！using known theory to present unknown situation,for example we have learned the sum of x divided by n! is e to the power of x,then lnx is e to the power of lnx which means x,that simplified this question a lot! I 've learned a lot via watching your video!

[@SatyaVenugopal]

This is glorious!

[@monishrules6580]

Thats such a cool question

[@ABruckner8]

My first instinct was "this probably simplifies to some kind of summation involving e," but I have no clue how to get there, lol.

[@freedomofmusic2112]

I'm a math tutor at my college, I gotta practice that two pen technique! I love using different colors when I'm helping students.

[@octopuskeng]

Amazing !

[@haaansolo8568]

This was a fun one!

[@neypaz8054]

I was missing your videos. =)

[@tomodachi1644]

Crazy how I know all the tools he used and still I coudn't solved it myself.

[@ahmed_hydrogen863]

Bro didn't know how to solve the integration He is the integration

[@rehmatullahkhan8969]

Good job sir black 👍

[@courageouscuber9278]

This is such an easy solution when you see someone else do it, but I would have never solved it by myself!

[@MrUberlyuber]

I love your content man. Now that I’ve solved it it seems almost trivial - of course it’s just an infinite series, :p

[@janitorvoniserlohn]

Let y = f(x), where f(x) is the integrand. Then after some tedious manipulation one has ln y =-1 + e^(ln x) = -1+ x So y = e^(x-1) The integration can be simplified as Integrate[e^t, {t, -1, 1}] = e^t, {t, -1, 1} = e - e^(-1) Hence, Integrate[f(x), {x, 0, 2}] = e - 1/e.

[@marcusviniciusdoprado7508]

What a beatiful integral and mathematics!

[@sunnyjha11]

I used to be able to solve questions of this difficulty level 10 years ago. I forgot most of the things and only remember some basic integral and derivatives formulas and concepts now :P Anyway this was cool to watch :)

[@mohammadalkousa2856]

Really Integration Bee very interesting. Thanks for video. For MIT integration bee recently it was published a book with solutions to problems of Qualifying Tests from 2010 to 2023

[@rybosny]

One of the best integrals I've ever seen. :)

[@guy_with_infinite_power]

It means you have not even seen 1% of the integral problems...

[@somebodysomewhere4670]

You could also use expansion of taylor series of e^x (its n-th derivative is always e^x) so that sum of those powers of x equals (x-1)/lnx and then substitute it back into the integral. Thats how i got it. Sorry for my english.

[@manojsurya1005]

That was a beautiful solution,clever

[@faheem5600]

Sir I m ur die hard fan Really amazing content delivered

[@andreh75]

You are the best! 🔥🔥🔥

[@Bayerwaldler]

So good! 😁

[@AayushSrivastava0307]

pretty easy ngl did it as soon as i thought of writing the radicals in terms of powers and the x^a must been some series expansion :)

[@YakshBariya]

Oh great question, not that difficult btw. I thought it would require some sort of wizardry, but when I wrote the question and simplified to the power of x, I quickly saw e^ln x, just needed to multiply and divide by ln x and create the e^x series, then it was pretty simple :) Once again, a great and simple question which emphasizes on finding patterns

[@pepe60735]

Fun. watching these videos make you learn methods to solve problems.

[@theupson]

if you convert each x (in the bases) to e^lnx, the series you get requires less finagling, as the power of lnx matches the index of the respective factorial

[@harstoft]

Calculus is so fascinating

[@alexander1989x]

Gotta love when things cancel out and you get a neat and manageable solution.

[@brazenzebra]

Beautiful! So easy to follow along with your argument. But, to come up with it on my own? Sheesh, it would take me months, maybe infinite months!

[@rhversity5965]

Hello Professor. I applied to UC Berkeley this year so I might see you on campus next year if I get in. If I do I’ll definitely join the Berkeley math tournament.

[@blackpenredpen]

I don’t teach at UC Berkeley. I live in the SoCal area but I go back to UCB campus whenever I can 😃

[@rhversity5965]

@@blackpenredpen Oh. Where do you teach then?

[@BenDover69831]

@@rhversity5965 he teaches in kenya

[@spaghetti1383]

Integrals like this are rarely hard. They are just time consuming because they require a bit of algebra. Once you simplify the integrand by knowing a power series, there are no integration techniques required. This problem is more like a race than a puzzle. But that's what most bee integrals are. You can't put many genuinely difficult integrals in an integral bee because of the format.

[@nouhznibi2297]

Your are the best bro 👍

[@blackpenredpen]

You too

[@randyla6706]

To the folks who want a more compact form: 2sinh(1). The hyperbolic sin function.

[@DestroyerWolfFenrir]

Loved it

[@akashmallick25]

Sir math is love for you and me❤️

[@pawandeep1424]

Your are great sir ❤❤😊

[@luigigg1741]

Love even the Klein bottle behind you!

[@gatjuatwicteatriek4590]

You deserve 500m subscribers

[@SpoiledJQ]

At first when i saw the thumbnail, i was like wtf is this. Then i saw him sythesize its complexity. It made it easier to solve in my head. Good ol' Calc II

[@nishchayy]

Amazing, that was amazing

[@doma3554]

The way he speaks softly and calmly, and is reassuring, reminds me of Steve from Blues Clues, except with integrals. :B

[@michaellarson2184]

I solved it without watching the video! I feel so proud!

[@kpax9284]

Very interesting and not so difficult to solve if you know series. Honestly I haven’t expected to solve that without substitution or partial integration

[@dhruvbhola9073]

He is a jewel

[@UnrealMatter]

Please do the derivation of ln(-1)

[@ItsPungpond98]

Mom! A new bprp is here!

[@Peter_1986]

Integrating like a boss.

[@wyatt6721]

Take my 👍. Great job

[@exodus8213]

Hello...i was wondering if you would know how to figure out if a private key is odd or even regardless of range in secp256k1

[@champu823]

I Didn't expect to solve this myself in 5min lol

[@bekapis]

amazingly good

[@maddenom]

This was an easy problem, (no offense), but the first thought in my mind was trying through u-substitution but I guess it would be easier to use Taylor series for defining e I mean it was pretty tough for some speed solving like an integration bee, but problems like these make you know about the essence of series...

[@pthezeus]

Diretamente do Brasil

[@mathmagic8577]

How to find the sum of these series (Nc0)³+(Nc1)³+(Nc2)³+....+(NcN)³.is it possible?

[@phoenixspirit6014]

Goddamn was that a beauty of an equation

[@Master_mind__235]

In √-1 put e^iπ and take out the root of -1

[@progboo]

Brilliant

[@theelk801]

did this one in my head

[@matteocilla9482]

and it is possible to find a primitive of this function?

[@jatinkapoor15]

I did this on my own just messed up a lil in the last . yay

[@aidenvilla7013]

I didn’t understand any of that but it was entertaining

[@Arthur_06]

Making of the thumbnail........this guy....give me that fucking medal 🏅

[@user-sh8hs1ij2j]

The braingasm at the end

[@swaggyseth1454]

HES POPPUNG OFF

[@agabe_8989]

beautiful

[@user-vt4bz2vl6j]

You could have started the power series from 0 as it already had the ln^0 x term, I feel it would have reduced some work. Amazing video as always...

[@ThAlEdison]

I needed a u substitution, e^u=x. To see the pattern, but when I got to e^(e^u+u-1)du, I converted back to the x world.

[@dante9632]

Sir how long did it take for you to solve this integral? Few minutes?

[@davidbean6973]

First time I’ve seen x to the power of a series! Lol

[@pdjibril]

Fantastic

[@CravenMaaaaaaaaa]

this dude can quick-select between a red and blue whiteboard marker

[@stringtheory5892]

Integral from 0 to 1 f(x)=2^[(logx)] Where [ . ] denotes the greatest integer function. Note: it is logx with base 2.

[@youngmathematician9154]

Here's how I did it: 1. Make the substitution u=log_2(x). This gives x=2^u => dx=2^u*ln(u)du. The bounds become -inf to 0. Our integral then becomes Integral(-inf to 0)(2^[u]*2^u*ln(2))du=ln(2)*Integral(-inf to 0)(2^[u]*2^u)du 2. Rewrite the integral as an infinite sum of smaller integrals. We are going to do so the following way: ln(2)*(Integral(-1 to 0)(2^[u]*2^u)du+Integral(-2 to -1)(2^[u]*2^u)du+Integral(-3 to -2)(2^[u]*2^u)du+…) =ln(2)*Sum(k=-inf to -1)(Integral(k to k+1)(2^[u]*2^u)du). The key insight here lies in the definition of the greatest integer function: for some integer k, [x]=k iff k