Solving ALL the integrals from the 2024 MIT integration bee finals
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- Опубликовано: 30 сен 2024
- Complex analysis lectures:
• Complex Analysis Lectures
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20:45 there should be a negative sign for the last integral so the last cot term is negative. Here are the timestamps:
0:10 Problem 1
6:09 Problem 2
17:37 Problem 3
21:47 Problem 4
26:55 Problem 5
Thanks and enjoy the video.
Master can you make a video soon for the proof of anti-beta function which you wrote int 0to∞ x^s-1 / 1-x^k = pi/k cot s pi/k
Your fan request please 🙌🙏
I did not get the same answer as you for the fourth integral. I supposed I must be making a mistake. To check I by put the integral into Wolfram Alpha ( no easy matter). To my surprise it gave an answer 0f 2.39027. This is virtually the same as the answer I got which was 4pi sin(3pi/5)/5. I did the problem starting in the same way as you . But them I observed that the integrand has a removeable singularity at 1 and two simple poles in the upper half plane. I closed the contour appropriated and then just used easy methods to find the residues. May be you can comment.
@@tomasstride9590 I forgot the negative sign for the 4th integral so that mistake is on me. I agree with the contour integration method that you've employed and if you're answer isn't the same then it must have been some minor error.
@@maths_505I accept there is a missing negative sign.. but I want proof for that equation master.. that [ pi / k cot ( pi s / k) ] none of the RUclips videos have proof for it... hoping you'll make a video soon for that proof 🙏🙌
@@joelchristophr3741 bro could you please stop with the master thing 😂 I mean it's a cool joke but I just like being called bro alot better 😂. I'll make a write up for it on my Instagram. See you in a few hours with a new video.
"ima have to differentiate the fuck out of this thing" is a hell of a line.
22:53
Happy integrating guys I need to learn physics for my exams
good luck....physics>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>everything else(literally)
How did it go? physics sucks ass
@@Physicsbbc Damn you got slandered
Imagine this IS your exam 😵💫
I have better explanation of 5-th integral solution:
Basically we just need to switch to binary number base, where 1/2 = 0.1, 1/4 = 0.01 and so on. It's very convenient for this problem because multiplying by powers of 2 just shifts all the digits in decimal part. As you mentioned in video, the function takes value = 1 for each x starts with 0.00..., for value = 1/2 x should start from 0.100...
And it's easy to formulate the general rule: the value of integrand equals 1/2^n iff (n+1)-th and (n+2)-th digits of decimal part of x are zeroes, and no two consecutive zeroes appear in positions earlier. This is quite standard combinatorics problem, and we come that count of combinations of n binary digits with no two consecutive zeroes equals to Fib[n+1]. We should also take into account that n-th digit must be 1, and the rest of calculations are basically the same you did in the video, and the answer is sum of Fib[n] / 4^n for all natural n values.
Also in final calculations of given series instead of Binet's formula you could simply use Fibonacci's generating function by substituting 1/4 there. I guess MIT contest finalists must use this formula which helped them to solve this problem in a fast manner)
7:00 - What could be better than watching an integration being done where the person in charge of the solution has a sense of humor. Mathematics skills combined with showmanship.
A ThreeBuleOneBrown video on your methodology for that horrendous last integral would be exquisite! I've stared at it for hours and spent 8 sides of A4 on it, whatever trick you need to know to do that in under 5 minutes must be pure madness.
I think problems with floors of 2^n can be done with some binary representation wizardry on the variable 2^n x. Then integrals like that should turn into a sum of fractions, but computer science never interested me lmao. I'll stick with multivariable calculus and differential geometry that's a breeze compared to that integral.
Serious respect to @maths-505 as always, awesome video!
I'd vote for the tie-breaker integrals. Nicely done.
I’ve only taken like half of calc 1 but I’m having a good time
4:10 there should be a e^(x/2) before 4(......)^(-1/3) right?
That's what I thought too! If not, then it doesn't make sense to think about it as the derivative of a product
This isn’t even calculus anymore
Where did the exponential go at 5:21 ?
I think he just forgot to put the exponential because if you expand out the numerator of the yellow integral, e^x/2 should have been distributed to the 4 as well. So it’s 4e^x/2.
25:44
I really love this solution but for the equation u^3-3u-2=0, the root u=-1 also works, why choose 2 over that?
also I guess a broader question, what do you do in these situations generally where the substitution is not a bijection
thanks
The integrand is non negative on the interval of integration.
Useful problems and fascinating solutions. Thank you.
the last integrate,suppose the total value is T,then integrating from 0 to 1/4 is 1/4,integrating from 1/4 to 1/2 is T/16,integrating from 1/2 to 1 is T/4,then you can solve T is 4/11
i ddnt get you, could you explain more briefly
Wow, this is actually so elegant
THE MOST AWAITED VIDEO, I kept patience and didnt watch any other solutions coz i only needed your solutions!
SUIIIIIIIIIIIIIIIIII
Watch mine solution too plz
@@maths_505 messi better
Messi better
The fibonacci series at the end can also be derived using its recursion relation properties.
If f(x) = Σ(k=0,∞)F(k)x^x
(1-x-x^2)f(x) = Σ(k=0,∞)F(k)x^k(1-x-x^2)
= Σ(k=0,∞)F(k)x^k - Σ(k=1,∞)F(k-1)x^k - Σ(k=2,∞)F(k-2)x^k
= x +Σ(k=2,∞)(F(k) - F(k-1) - F(k-2))x^k
= x
f(x) = x/(1-x-x^2)
21:48 I've actually wanted to use Cardano for an integral since a long time ago 🤣
Most simple solution of problem 4 I found:
Let f(x) be integrand. Then indefinite integral of f(x) is F(x) = 2(xf(x) + log|f(x)|)/3 + C. Therefore, I = F(1) - F(-1/3) = 14/9 + 2log(2)/3.
I am curious what is the drawing software? That looks very clean!
easiest calc 2 exam
Im gonna enjoy it
In the 4th problem with cardano, when you are finding limits, for x = +1, the equation is u^3 - 3u -2 = 0, for which one solution is u = 2, but another solution is u = -1. May be expand on this a bit? may be by going to the original equation for u in terms of x
u cant be negative for the given limits
The integrand can't be negative given the interval of integration. Oh someone's already answered 😂 cool.
@@maths_505is it possible to have the method fail cause of multiple valid solutions? 🤔
@@nicolastorres147 nah we can just split up the integral into 2 or more and apply the appropriate limits.
ayo i subscribed brother. Love from India. Love your videos too man
Love from right across the border homie
I always love ur videos because of the dark background 😂 I use dark mode in every website and app😂😂
Math on dark mode 😎
Why didn't you participate in bee finals.
I'm 110% guaranteed that you will win😅
Btw these integrals are "fire emoji"
I would've gotten atleast 2 and 4 correct within 4 minutes and that would've sealed the deal 🤣
Did you solve the first two problems with Feynman's technique?
bit late but at 3:56 why wasn't the e^(x/2) expanded out to both, why just for (3cosx+4sinx)^(2/3)
Hey | Alpha|
How do I learn this wizardry? Any textbook recriminations??🙏
I just finished studying indefinite integration for the first time and d my friend ( also in similar situation) gave me the the first question to integrate since i saw a term of exponential in numerator i thought this integral would take the from of e^x (f(x)+f'(x)) but nothing i did made any sense...I have to say integration is pretty . Thanks for the solution !
In the words of peter drury : Wonderful , Wonderful , Wonderful.......
for the first integral, isnt the second half missing an e term? or is there supposed to be brackets around the whole thing
3:39 you missed the brackets i got confused for like a minute but then realised that 😅
At 25:42 , the value of u could have been -1 as well as it is satisfying u³-3u-2=0. You chose 2, which also satisfies the equation. Question is how do we know which one to chose?
Same question here
Because he lets u as the sum of two cube root which won't be negative in this situation
@@wilsonlu168 why can't sum of two cube roots can't be negative?
(-1)³+(-2)³= -9
Though I forgot the content of the video as you can see I've made that comment a month ago.
@@shubhankardatta2437 no mind
the sum of two cube roots can be negative as you wrote
but in this situation the integrand is non negative here
If making the value of u negative here, ln(u) afterwards can't be performed
I love it
I do not have the requisite knowledge for this
Where did you pick up a Swedish accent? 😀
What writing board are you using for this one?
What resources whould you recommend for integration
Love your content all my salutations from Morocco
🤯🤯🤯excelente esto es lo mejor 😁
Classic Maths 505 content
yeah they aint doing allat in 5 min per problem
😭 literally tells us its drastic 1:37
Like for your first fibonacci boy of the channel!🎉
bro , you had to mention that n is a whole number , i was solving it thinking it is to a non negative number
anyways , great video
Hey friend 😊. I solved all of them myself, except for the last yellow one. That required a hint
How did you get so good at math
I'm not exactly good just really persistent.
Are these advanced integration techniques?
That is impressive.
At 25:29 you say that when x goes to 1 u goes to 2. But why not to -1 for example. The cubic equation has more than 1 solution. How do you know which one to choose?
The integrand isn't negative on the interval of integration.
What if we integrated beyond 1 where we would get a real number as a sum of conplex conjugate cube roots? 🤔
The cube roots of complex conjugates need not be conjugates of each other so I don't that would be viable.
Here is the timestamp :
0:10 Problem 1
6:09 Problem 2
17:37 Problem 3
21:47 Problem 4
26:55 Problem 5
Nice video.👍
Wow,well done!
41:45 maths 505 vid? Yes please!!
Where can I find proofs to formulae at 19:40?
On my Instagram soon
Best integrator 💪💪
I waited for this!!!
My bro is on fire!!!
bless you
Love this
Thank you
--at the first problem: using the function with its derivative to find A and B;is it somekind of integration trick?
More of an educated guess
I'm a moron, I know it. But sometimes I wonder, with all this focus on weird integrals that don't really seem to tell a story, do these kids know numbers? How many of these kids looked at a set of normalized whole numbers from negative max to positive max and associate the sphere with R^3? How many of them know that you can color those spheres with quaternions using single input or 3 input values? How many of them know there are strong associations between algebra and trigonometry? I know this stuff and it's not because of higher education. I just explored a bit on my own. I have grade 12. And I will never go beyond that.
In 2nd question,
You made it very complex
Just use
Numerator: x + ln(2-e⁻ˣ)
Denominator: eˣ - 1
Then substitute t = e⁻ˣ
Then integrate t=0 to 1
Numerator: -ln(t) + ln(2 - t)
Denominator: 1 - t
Now replace t→1 - t
Numerator: -ln(1 - t) + ln(1 + t)
Denominator: t
Now it is ς(2) + η(2)
Simple
Btw lots of respect from Indian CSE undergrad 😅
I just couldn't resist a Feynman's trick approach 😂 your solution is excellent.
@@maths_505
Then you should use this complexity for third question, just by the use of complex analysis and contour integration.😂😂
On residues at 72° and 144°
e^(2iπ/5) and e^(4iπ/5)
Then answer will be just
(4π/5)sin(2π/5)
Thankyou i was searching for it 😭