Solving all the integrals from the 2023 MIT integration bee finals

Sit back, relax and enjoy the wild ride of evaluating the beastly integrals from the 2023 finals.
Thank you Myers for the wonderful solution development for problem 2.
Thank you Poro :D for the timestamps:
Q1 : 0:11
Q2 : 4:49
Q3 : 12:26
Q4 : 21:24

[@maths_505]

I made a mistake while editing the video: For the 2nd integral, use double angle formulae for sin2x, sin6x, sin10x and sin30x to get the integral on line 2 at the 4:51 mark. The solution development for this integral is credited to my friend Myers Solutions for the 2024 finals: kzbin.info/www/bejne/h5KsZGuOhNikgpIsi=UBudXz7fv9LOAS8z

[@jackychanmaths]

Where's the floor function in Q4?

[@maths_505]

Left as an exercise for the reader. Its actually not that hard.

[@kurrurin]

i,.

[@jackegan9947]

🙄

[@sharp_object]

i like your funny words magic man

[@cupidstunt22]

You lost me at Ladies and Gentlemen...

[@youtubecommentrater3379]

Funniest thing I've read 2023, so far 😂😂

[@marklo6283]

Wait you thought women cant do the math bee? Sexist much 🤨

[@ibrahimtemiza8947]

Why hhh?

[@nepatriots11]

🤣🤣🤣

[@lucianopollicino]

​@@user-ws9dl9jx7fNani?

[@eobardthawne19]

The fact you are going through so much effort to post out these high quality videos is insane! Keep up the good work and ima bet you are gonna grow faster than Ray's function ( well not actually tho lol that wud blow up KZbin servers)

[@Kuzmo]

whats rays function

[@NoceurXeno]

Yeah, what's ray's function????

[@eobardthawne19]

@@NoceurXeno rayo's function

[@polyreza]

I really enjoyed solving the last integral, the monster one😂 It was something completely new to me. Thank you

[@SunnyRosy-uk1ch]

Id say the last one is the simplest to solve at first glance for MIT level students in an exam as once you get that it's an infinite GP, then the question gets solved in like at max 4-5 steps

[@two697]

I'm surprised they were able to answer the last one but none of the first 4 integrals. I would say that the last integral was the hardest

[@jhadhiraj147]

Last one is straight forward bro. 3:08

[@oumguptahello]

@@jhadhiraj147 can you elaborate it please?

[@Pal_yt]

If you're Asian then you could arrive the answer without picking the pen.

[@__Hanasei__Levinus__]

@@Pal_yt hey hey, I know it's a joke, but tone down the american racism a little bit... We don't need anymore people coming from asia have societal expectations imbibing us with this silly pressure in regards to mathematics, lolol. ...seriously, I don't want strangers to expect me to solve these kinds of problems, and then be laughed at, and then looked at, as someone who is silly and weird....

[@clvsidy]

@@Pal_ytLMAO

[@manstuckinabox3679]

Bro the Square decomposition was a new thing to me lol! and that last one was just mind boggling!

[@johnchessant3012]

Very nice video! Just a note, Q4 required the contestants to find the exact value of the floor of the answer, so you'd need to do a series expansion to a few terms and check that your residual is less than 1. Elementary but very tedious haha

[@mickodillon1480]

Indeed Sir

[@martinepstein9826]

At 10:07 here is a quick way to see int_[0 to pi] 4 cos^2(8x) dx = 2pi. By periodicity it's clear that if we replace cos with sin we get the same answer. Now if we add we get 2*answer = int_[0 to pi] 4(cos^2(8x) + sin^2(8x)) dx = int_[0 to pi] 4 dx = 4pi Therefore the answer is 2pi.

[@petrie911]

For problem 2, it's easier if you substitute u = 2x so that the integral covers a full period of cosine. Then once you have your sum you can use the orthogonality of Fourier series terms. It also saves a lot of extra 2s in the intermediate calculations.

[@tikarai4060]

Interestingly solved! Great video. My favorite ones are 2 and 5. Thanks a lot.

[@venkybabu8140]

All integrals and differentiation is like the gearing systems of a car or watches. Functional transforms and substitution of limits is like the size of the gears and there number of teeth. When limits are changed from one to other it is somewhat like changed number of teeth on the gears and clutches. Functional transforms are like sine cosine etc. the rate of change of functional parameters. Maths is somewhat like that. A gearing systems with dimensions. FofF is dimensions switch. Or base or radical changes.

[@IroineGrandison]

Even if you give me a year to solve the last one I would never be able to solve it, thanks a lot mate hope your channel grows more, keep it up

[@mukaddastaj5223]

Man, this is dope! It was so consice, with no extra words and the problems were beautiful😍 i'm still in HS, so i'm not really familiar with gamma and beta functions, but i sure gonna do some research now🤭😃

[@avyakthaachar2.718]

Thank you so much for this video ❤

[@wynautvideos4263]

C’est vraiment incroyable que ils peuvent resoudre celles problèmes aux quatre minutes.

[@Animal-yb1rr]

Only bee that I integrated with was a bee that stung me

[@maths_505]

😂😂😂

[@Db--jt7bt]

There should be a side contest for people integrating with approximations like Riemann Sums and Simpson’s Method.

[@piyushsingh178]

Good effort man🎉 In Q3, you'd need to go further ... You'd have to do infinite sum form of log(1+x) upto 3 terms and that will give you the final answer

[@MONDLIMTHOMBENI-tt2vl]

This are long and 😂 what in 4 mins it's what those guys were doing in their heads

[@trimmy8461]

I'm pretty proud that you only lost me at the beta and gamma functions

[@theelk801]

I would’ve done residue theorem for the second one but I never pass up an opportunity for that

[@AbdullahPunctureWale-II]

Loved the first one... going to watch the rest... ❤❤

[@NaitikNaitikGupta-f9d]

Honestly problem 6 was done by me but in 15 minutes

[@noelani976]

FYI, problem 4 had the whole integral under a FLOOR FUNCTION.

[@Youtuber_YusukeFromGermany]

I want a professor like you to make math fun again in university

[@yuseifudo6075]

Hello fellow Yu-Gi-Oh fan

[@Youtuber_YusukeFromGermany]

@@yuseifudo6075 I see you are a man of culture as well

[@realleglesslizard]

as someone who thought i was good at calc bc this year, i have no idea wtf just happened

[@sangeetatiwari6007]

11th grader me watching it and having absolutely no idea about whatever is going on😂😂😂

[@ttrss]

So many times in this video, I pause for a few minutes and then say oh I'm stupid, then continue watching

[@racquelsabesaje4562]

/ pi/2 0 3/tanx/(sinx+cos x)2 dx

[@alejandrojimenez108]

So are you not going to do the fourth problem with the floor function? The floor function is what actually makes it difficult

[@maths_505]

Why would we skip the floor function bit? That's the whole point of the integral!

[@two697]

​@@maths_505 he's talking about the question where you let u=x^10. In the original problem, there was a floor around the entire integral

[@maths_505]

Oh that... The floor function applies to the solution which makes life much easier.... It's the 5th problem where we're actually integrating using the properties of the floor function.

[@alejandrojimenez108]

@@maths_505Yes, but how do you evaluate the floor of the answer for that problem?

[@ТөлеубекЕркенже]

Would you mind sharing programm you using to draw your solutions? I would appreciate that. By the way, very cool vid, keep it up man!💯

[@maths_505]

Samsung notes on my S6 tab.

[@funnyguydragon]

Oh god i need to study these stuff this year 😭

[@RoyMustang.]

Bruh I don't understand but I'll one day sure 😭

[@pacman7168]

WTF! Never has seen this type of integrals!!! 😅😅😅 All the days learn new things. Cheers from Caracas Venezuela

[@barancbik]

This is so easy they should put me to top of the MIT

[@gel2709]

3:27 As a russian, I understand. Sorry about that 😓

[@Nifton]

Мне кажется, что здесь это ни при чëм 🙂

[@gel2709]

​@@Nifton Когда вы написали этот комментарий я тоже как-то осознал, что это скорее всего здесь ни при чём. Я не знаю, я постоянно думаю об этой теме, вот и вылезло. И мне наверное это напомнило ситуацию: Я работаю репетитором по математике и у меня когда началась война один ученик отказывался использовать в задачах вообще букву Z. Через пару недель это прошло конечно, но как-то это воспоминание сразу в голову ударило, когда я смотрел это видео. P.S. За Элейну на аватарке однозначно лайк

[@PinoySarisariTV]

I think in 14:03 it should be 2x^2 and not 2x

[@PinoySarisariTV]

lmao, i should have watch first before leaving this comment. haha. 14:35 it is

[@TeoCurcudel-yv6ws]

Great video! What app do you use to write these?

[@anvimishra9702]

what jee mains expects from us

[@gursimarmiglani9143]

how tf they supposed to solve the last integral in 4 minutes lmfoa

[@bobbobert9379]

"Oh yeah thats just the beta function" Me, a math major: 👁👄👁

[@haroonsagheer5050]

hello, which device and app you are using for wiritng maths so smoothly? please tell me for i need it for maths classes

[@SK_LynIx.7k9546]

is there no other shorter method for last question? And how did Luke did it in less than 4 mins??

[@maths_505]

Luke is a genius I am not a genius That's how EXACTLY how

[@SK_LynIx.7k9546]

@@maths_505 That's not true, if one works really hard can reach anyone's level....don't degrade yourself by saying that. Is there anyway that we get to know how did Luke solve it ?

[@par22]

Kinda skipped some steps at the start of 2nd problem 4:49 lol

[@maths_505]

Oh damn Thanks for pointing that out mate

[@par22]

@@maths_505 np that's a lot to keep track of lol. good job on the marathon, really impressed with it all, especially that last one

[@maths_505]

@@par22 thanks mate For the record going from step 1 to step 2 uses the double angle formula for the sine function

[@tedjohnson64]

Very interesting!

[@adraino7345]

KZbin algorithm continues to surprise me

[@artemis_furrson]

As soon as people pull up some bs obscure functions to solve integrals, I'm out. Most I can do is integrate by parts and substitution.

[@racquelsabesaje4562]

I=7 pi

[@ehsOFspades]

What level of calculus is this ?

[@domc3743]

The way luke just stands there scribbling the odd thing on the board every now and then and doing everything mentally is breathtaking in a way, what a talent. Appreciate you solving these integrals for our pleasure

[@analysis726]

i could’ve never figured out factoring out cos^2 from (sinx+cosx)^2 in the first problem in order to set up a u-substitution. it’s such a nice, simple solution that requires just a bit of outside the box thinking to find.

[@kawamann1234]

i still dont get how this factorization works out

[@kawamann1234]

ah nevermind i got it ...haha

[@familyfamily6037]

@@kawamann1234 how does it work

[@kawamann1234]

when you are going to factor out the cosine, you need to take care about the power of 2 of the whole expression. Try writing it simpler and add more steps by writing it as whole like (sinx+cosx)(sinx+cosx). When i did that i could see it immediately. But just in case: This leads to cosx(sinx/cosx + 1)*cosx(sinx/cosx + 1) then you see its the same expression twice and there you go. @@familyfamily6037

[@muditchaturvedi5276]

Exactly why to go for beta and gamma fns. ......

[@epikherolol8189]

Ah this is why I couldn't solve the first one no matter how much I tried. I'm a highschool student and here in my country they don't teach u gamma functions and beta stuffs.

[@IndranilBiswas_]

What beautiful integrals!! Calculus is where trigonometry really shines as an elegant and versatile subject. Who would have thought you could derive so much utility from Pythagoras's theorem!

[@Wosndolousn]

The fuck you guys talking about?

[@jasiek1405]

I understood absolutely nothing, I don't even know what an integral is, yet I enjoyed this video so much

[@maths_505]

😂😂😂....I'm glad you had fun

[@Ligatmarping]

Well, this are certainly NOT the first integrals you should try to solve when you learn what they are xd. They are beautiful and the video is very good, but there are a lot of simpler ones to practice.

[@michaelblankenau3129]

Same here . I keep expecting that through osmosis I’ll be able to understand some of it… but it never happens

[@yuyuvybz]

How good is your integration now 👀 I want to see how good I'll be in 1 yr too.

[@TUMATATAN]

Wow, I can attest there were SOME words in English...I believe in the beginning he said "Ladies and Gentlemen..."

[@LeonidArgail]

How is it possible to solve fifth problem in 4 minutes? Unless you know this and remember answer.

[@anonimus7456]

what is this language he’s speaking and writing 💀

[@suvosengupta4657]

the last integral is something of its own😵‍💫did you have the same reaction while solving🤣:D

[@maths_505]

I definitely enjoyed it

[@jose4877]

No way that you can solve the second integral like that in under 4 minutes. You even cheated using wolfram alpha. There has to be better or more clever solutions than the ones shown here.

[@kutubkhanbhatiya4573]

Ohh man I was searching for the solutions of these integrals for a long time. Thankss

[@hozinryu]

In the second integration, I remembered the orthogonality of the cosine function, being able to effectively cancel out a few terms!!

[@oumguptahello]

for the last integral how's the winner able to get to answer really quick without moving much of a hand . Apparently I struggled to assume what to do in it so I took this to my professor he was able to give me the range of this question via sandwich theorem but didn't able to land to on answer.

[@wondrousparrot2815]

Idk why I’m watching this as I’m in precalc but this is some dark magic holy shit.

[@Ligatmarping]

Great video! My favourites are also the third one and the fifth one (in the order you make them on the video). This are the ones with the really nice ideas in the solution. The fifth one's idea of comparing to the integral from 0 to 1/2 is kinda natural thanks to the 2^n in the sum. The third one is the one I wouldn't have a big hope of getting something nice while trying to get a and b haha, but it works.

[@matei_woold_wewu]

Summation inside of an integral

[@insouciantFox]

I might have used Euler substitution at the sqrt(x²±x+1). It happens so rarely but is so satisfying

[@mathcanbeeasy]

14 minutes for the last one, given that you had already solved it and presented it as quickly as possible. How the hell could anyone have solved it in 4 minutes?

[@joseph-jg2ie]

Where does one learn these unique methods of integration? Im now wondering if I purchased the wrong calc textbook, not sure I remember seeing these tricks... (If anyone has any recommendations on how I can level up my game pls lmk)

[@bhattbhavik9709]

You can solve JEE PYQs.

[@sanchitlodha5770]

in the third question when we get 2rootof ab we could just have added and subtracted x^2 in the root of the rhs and then use a^2-b^2=(a-b)(a+b) to factorize and take a 2 outside the root to immediately find the and b quicker is what i think

[@alien77333]

Hey can you please tell me what are the uses of such hard integrals in our life? Integrals are useful for finding areas and for physics calculations but these long trigonometric integrals 😐.

[@maths_505]

You'll find lots of tough integrals in higher physics and applied math....these are just like practice problems to help think

[@alien77333]

@@maths_505 After watching this video I had a great discovery. Big enlightenment. I can safely say that I've come to the genius conclusion that I understood nothing.

[@maths_505]

@@alien77333 😂😂😂😂😂😂😂

[@DiegoTorres-iw5kn]

WHAT KIND OF WAY IS THAT TO WRITE π

[@himanshu934]

at 15:05 why didnt you take a - b = +x and a- b = -x as two different cases?

[@notaslave9628]

Maybe we imply that x>0

[@WforWill]

I watched this right after taking Calc AB. Im scared 😂

[@PritamDa-q4u]

Which software are you using, sir ?

[@maths_505]

It's just my note phone's default software

[@lyrocon5951]

Which program do you used to write on the phone? (i mean the black board)

[@webber9449]

crazy how the mit kids do it in like 4 minutes

[@sickomode6440]

Literally me during the entire video 8:28

[@EricraftDeeb]

I wish I could understand at least 1% of what you did

[@scratchbeginners4541]

I really want to learn those extra techniques for integration(such as Gamma function(????) and Beta Function(????)). Could you recommand any books/videos to learn those life-easing techniques?

[@jondon3925]

my coaching is dumb af the 2nd question they repeated in our weekend exams 😑

[@SK_LynIx.7k9546]

you from india?

[@jondon3925]

@@SK_LynIx.7k9546 yes mate

[@nafio2206]

Guys I made it to 1:36 yay. No further tho

[@oo_rahbel_oo]

21:17: Just for the record: The denominator of the logarithm must be the root of 3 instead of 3. Regarding to the last problem: This is the first integral where we know that there is a (closed) solution and mathematica is biting its teeth on it (at least I didn't manage to find any solution using wolfram). Are humans perhaps smarter in the end? Apparently they are at MIT. 😊

[@abi3135]

Isn't the time limit for integration bee finals about four minutes per question? Damn expecting to solve 4:58 in about that time is just cruel

[@maths_505]

1. I'm not as smart as the bois at MIT 2. They don't have to explain anything while solving😂

[@two697]

4 minutes is definitely unreasonable for all of them except maybe 1 or 4. Only one of the finalists was able to answer a single integral

[@abi3135]

@@two697 3rd one's possible within 4 minutes i think, if you know the de-nesting formula beforehand instead of deriving it √(a ± √b) = √((a + √d)/2) ± √((a - √d)/2), where d = a² - b²

[@two697]

​@Abi solving the integral isn't that bad but getting it into the final form is quite cumbersome. Either way, you've basically got to know how solve the integral at first glance as you haven't got time to explore more than one method.

[@poojankhunt8257]

I had already solved 3 of them before seeing this video, still 2 left to solve......

[@김현규-e4v]

Fantastic video: It definitely makes me study much harder than I used to-

[@smoothsentient]

Note the following x^20 - 48x^10+575=(x^10-24)^2-1, so if u = x^10-24 you could make substitutions faster

[@wryanihad]

Integrate 1/(x²-1) =arctanh(x) Doesnt need partial fraction to become Ln Do you agree with me?

[@Aman-cg2yt]

This integration from Beta and gamma function and good question to explain all how to gama function work 4:52

[@jyotiradityashukla4569]

Damn!! I was able to answer some of them.

[@worldblazin4209]

Woah this very well amazing. Thanks for MIT Integration this year and solving all problems. 🥰🥰😝

[@marciamarquene5753]

Si for assim assim mesmo né minha vida sempre vcs vão fazer um pedido para mim também e muito obrigado pela informação do coração e no fundo do céu no médico amanhã às dia todo e e e

[@marciamarquene5753]

Só r w It só r w It só no centro da minha vó hoje né em em frente ao mercado livre e muito difícil né minha vó e um beijo e um pedido do nada ele falou para todos os momentos do

[@sophiaisabelle01]

This is more complicated than all the problems and issues that I have to deal with.

[@MrGreenWhiteRedTulip]

Thank you for this video, amazing content! My integration isn’t good enough to follow the bee itself but this kind of video really helps me enjoy it! You have earned a new subscriber :D

[@marciamarquene5753]

Do jeito r w It e muito difícil né minha vida sempre e sempre e sempre e sempre e sempre vcs vão fazer um beijo e e e e e muito melhor

[@marciamarquene5753]

Si for fazer nada e nem um beijo no rosto do homem da minha vó hoje e e e muito melhor do Brasil no centro do Brasil e do Brasil no médico e e e e e e e e e muito melhor agora da

[@mohammadalkousa2856]

Many thanks, useful video! Also for the qualifying tests of the MIT Integration Bee. Recently it was published a book with a title (MIT Integration Bee :Solutions of Qualifying Tests from 2010 to 2023 ), it is very useful

[@louisl7245]

May I know where can I download that book? Thanks

[@himanshu934]

and for problem 4 the person says the answer isnt divisible by 5 I got the same ans as you but both our answers are divisible by 5

[@marciamarquene5753]

Do nada ela falou para mim também e muito difícil né em casa e e e e muito melhor do dia todo dia inteiro e e e muito difícil fazer um pedido para mim também

[@SaidThoughts]

"pretty much clear as day" >.>

[@marciamarquene5753]

Du nada e e muito melhor agora da escola e muito difícil fazer um beijo e e e muito melhor agora no centro do rio em em