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_]

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:

[cupidstunt]

You lost me at Ladies and Gentlemen...

[domc]

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

[analysis]

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.

[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!

[jasiek]

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

[eobardthawne]

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 YouTube servers)

[two]

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

[polyreza]

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

[manstuckinabox]

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

[johnchessant]

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

[kutubkhanbhatiya]

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!!

[martinepstein]

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.

[Jozehkmz]

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!

[petrie]

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.

[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

[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.

[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

[mohammadalkousa]

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