Solving all the integrals from the 2023 MIT integration bee finals
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- Опубліковано 27 бер 2023
- 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
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:
ua-cam.com/video/QaI38XOsqL0/v-deo.htmlsi=UBudXz7fv9LOAS8z
Where's the floor function in Q4?
Left as an exercise for the reader. Its actually not that hard.
i,.
🙄
i like your funny words magic man
You lost me at Ladies and Gentlemen...
Funniest thing I've read 2023, so far 😂😂
Wait you thought women cant do the math bee? Sexist much 🤨
Why hhh?
🤣🤣🤣
@@user-ws9dl9jx7fNani?
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
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.
i still dont get how this factorization works out
ah nevermind i got it ...haha
@@kawamann1234 how does it work
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
Exactly why to go for beta and gamma fns. ......
Ohh man I was searching for the solutions of these integrals for a long time.
Thankss
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.
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!
The fuck you guys talking about?
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 UA-cam servers)
whats rays function
Yeah, what's ray's function????
@@NoceurXeno rayo's function
Bro the Square decomposition was a new thing to me lol! and that last one was just mind boggling!
I really enjoyed solving the last integral, the monster one😂 It was something completely new to me. Thank you
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
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
Thank you so much for this great video. I’m noting down all the techniques to practice them and trying them out on new math problems.
I really love your video and appreciate what you’re doing. I’m learning a lot from it.
Interestingly solved! Great video. My favorite ones are 2 and 5. Thanks a lot.
Thank you so much for this video ❤
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
Last one is straight forward bro. 3:08
@@jhadhiraj147 can you elaborate it please?
If you're Asian then you could arrive the answer without picking the pen.
@@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....
@@Pal_ytLMAO
Thank you sooooo much. Really appreciate
Woah this very well amazing. Thanks for MIT Integration this year and solving all problems. 🥰🥰😝
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.
kudos for the efforts!! i hve been finding the solutions to these probs but noone posted a video or something. Thanks
Fantastic video: It definitely makes me study much harder than I used to-
Amazing! The last Integral was cool.
Thank you so much for uploading this. Yesterday I found the exercises and I couldn't be calm until I found the solution. Now I can finally rest haha. Great job, keep on going!
In the second integration, I remembered the orthogonality of the cosine function, being able to effectively cancel out a few terms!!
That really was a wild ride
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
May I know where can I download that book? Thanks
I understood absolutely nothing, I don't even know what an integral is, yet I enjoyed this video so much
😂😂😂....I'm glad you had fun
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.
Same here . I keep expecting that through osmosis I’ll be able to understand some of it… but it never happens
How good is your integration now 👀
I want to see how good I'll be in 1 yr too.
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
Indeed Sir
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🤭😃
I might have used Euler substitution at the sqrt(x²±x+1). It happens so rarely but is so satisfying
This is beautiful and artistic
Loved the first one... going to watch the rest... ❤❤
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
The first good math video on youtube ❤❤
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.
Note the following x^20 - 48x^10+575=(x^10-24)^2-1, so if u = x^10-24 you could make substitutions faster
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.
Good job!
thank you boss
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
Awesome video. I'll note out that in the third question, a small simplification we can use is that (x+1)^2-(x+1)+ 1 = x^2+x+1 and shift the bounds and just do only 1 integral
so cool!!!
C’est vraiment incroyable que ils peuvent resoudre celles problèmes aux quatre minutes.
I would’ve done residue theorem for the second one but I never pass up an opportunity for that
AWESOME
you’re a wizard
This is crazy but fun.
The ability to integrate beyond target, is what separates an obtuse predator from an acute producer.
En el tercer ejercicio te faltó poner al tres dentro de la raíz cuadrada en el resultado final de la integral.
Very interesting!
There should be a side contest for people integrating with approximations like Riemann Sums and Simpson’s Method.
Idk why I’m watching this as I’m in precalc but this is some dark magic holy shit.
Im surprised I could follow all that. Nice job
ua-cam.com/users/shorts-5Rrl56dBJo?si=DXvyMY02wCR_8iE-
Here is my attempt to the squared summation integral:
1. Expand the squared summation and split into 2 parts, the squared terms and the cross product terms
2.For squared terms, resolve the integer function term by splitting the [0,1] interval into [i/2^n,(i+1)/2^n] for i=0,...,2^n-1
3.For cross product terms, resolve the integer function term by splitting the [0,1] interval into intervals such that int[(2^i)x] and int[(2^j)x] jumps over the consecutive integer pairs {0,0},{0,1}...,{0,2^(j-i)-1},...,{2^i-1,2^j-1}
4.Integrate the resolved constant functions term-wise and apply arithmetic & geometric series
Oh god i need to study these stuff this year 😭
Thank you. I didn't understand anything but this was fun
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.
He is the Formula King!!!
I really like 👍 solutions that integrate. It's not hard
just watched the first one. absolutely brutal
Destroyed that integral with beta supremacy 🔥🔥🔥
I'm pretty proud that you only lost me at the beta and gamma functions
I wish you were my best friend all throughout my math major.
Q3. Let, V m, n in R three functions are defined by
f(m, n) = integrate x * (m * x ^ (1 / n) - n * x ^ (1 / m)) ^ ((- (m + n))/(m + n + 2mn)) * (x ^ (1 / m) - x ^ (1 / n)) dx from 1 to 2 g_{1}(m, n) = f(m, 1) - f(2, n) And g_{2}(m, n) = f(1, n) - f(m, 2)
Then the value of [g_{1}(2, 3) + g_{2}(3, 4)] is:. ([.] Denotes the greatest integer function)
Great video! What app do you use to write these?
Damn!! I was able to answer some of them.
yooo MIT integration bee 2023 had some great questions
¿What is the name of the program you use to write and draw? Thank you in advance. Great video sir.
For the first problem you can simply sub z = u^1/3 and then solve it using elementary techniques!
How is it possible to solve fifth problem in 4 minutes? Unless you know this and remember answer.
Very impressive! Those pi:s though...
Those are hard!
the last integral is something of its own😵💫did you have the same reaction while solving🤣:D
I definitely enjoyed it
Summation inside of an integral
May I ask you what note-taking app are you using for? Thanks a lot!
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!💯
Samsung notes on my S6 tab.
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
This are long and 😂 what in 4 mins it's what those guys were doing in their heads
great video as always!
how did you factor out cos^2x from sinx+cosx?
It's (sinx + cosx)^2, so factoring out cosx from sinx+cosx inside the brackets, you get (cosx*((sinx/cosx) + 1))^2 = (cosx)^2*(sinx/cosx + 1)^2.
I had already solved 3 of them before seeing this video, still 2 left to solve......
Which program do you used to write on the phone? (i mean the black board)
So many times in this video, I pause for a few minutes and then say oh I'm stupid, then continue watching
When I was in college, this would have been great to attempt. (BS math from UCLA)
nice video
Wow, I can attest there were SOME words in English...I believe in the beginning he said "Ladies and Gentlemen..."
"pretty much clear as day" >.>
I watched this right after taking Calc AB. Im scared 😂
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?
Just commenting to boost the algorithm, nothing more to add. This was insane.
Thanks mate
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.
Question on the first problem you did but how were you able to factor out a (cos(x))^2?
Wouldn’t redistributing that create (sin(x))^2(cos(x))+cos(x)^2, assuming you include the cos(x) in the denominator of sin(x) to cancel out a cos(x)?
To add to that, how were you able to include a cos(x) in the denominator of sin(x)? just wondering where that came from.
Also how were you able to keep the contents in the denominator (tan(x)+1) squared despite factoring out cos(x)^2?
Generally speaking, I do see how the trig identity of sin/cos creates tan(x) and that 1/cos(x)^2 creates sec(x)^2 but I’m still a little confused on the other aforementioned parts.
Any answers would be greatly appreciated. Thank you!
Expand the (tan + 1)^2 and (sin + cos)^2 , you'll find the answer.
sin x + cos x = (cos x) (sin x / cos x + 1) = (cos x) (tan x + 1). Therefore, (sin x + cos x)^2 = (cos x)^2 (tan x + 1)^2
(sinx + cosx) = cosx(sinx/cosx + 1) =cosx(tanx + 1)....this implies that
(sinx + cosx)^2 = (cosx)^2(tanx + 1)^2
14:13 2x^2
In case anyone is wondering the integral of first function is (-1/2((((tan(x))^2/9)-1)^1/3)((tan(x))^2/9))-(((((tan(x))^2/9)-1)^1/3)/2((tan(x))^2/9))
FYI, problem 4 had the whole integral under a FLOOR FUNCTION.
Can you please let me know what app you are using for this explanation ?
Hey i tried to do the first one by substituting sinx and cosx in tanx/2 form and got pretty far. Im stuck on the last part. Can u cover it in your video?
hello, which device and app you are using for wiritng maths so smoothly? please tell me for i need it for maths classes
I did all my calc classes. Learnt every integration trick in the book bot hot damn these are difficult. I'm guessing preparing for these is a lot of learning standard integrals?
Only bee that I integrated with was a bee that stung me
😂😂😂
What is beta and gamma function first time heard about it. I never heard about it earlier
What program do u use to write ? Can you tell me 💯
Timestamps
Q1 : 0:11
Q2 : 4:49
Q3 : 12:26
Q4 : 21:24
Q5 : 25:05
The solution to the first problem demonstrates why I failed again and again via partial fraction stuff...