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: th-cam.com/video/QaI38XOsqL0/w-d-xo.htmlsi=UBudXz7fv9LOAS8z
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.
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
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 TH-cam servers)
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
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!
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.
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.
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.
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
@@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....
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.
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
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
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
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)
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.
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.
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?
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.
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.
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. 😊
25:02 Are you simply going to skip the floor function part of problem 4? The whole challenge is to find an integer solution to the floor of the answer. At 12:02 Small mistake that does not affect the result, but you forgot a factor of 1/2 from the product to sum identity. In the end, the result is still 0. You could’ve avoided a lot of the mess in problem 5 by simply cancelling the 2^(n-1) and the 2^n at 36:09, leaving 1/2, instead of expanding and inducing errors.
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?
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!
@@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
@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.
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.
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)
th-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
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
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 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.
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!
@@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
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:
th-cam.com/video/QaI38XOsqL0/w-d-xo.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
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. ......
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?
Ohh man I was searching for the solutions of these integrals for a long time.
Thankss
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 TH-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
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?
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.
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.
In the second integration, I remembered the orthogonality of the cosine function, being able to effectively cancel out a few terms!!
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.
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
Wow, I can attest there were SOME words in English...I believe in the beginning he said "Ladies and Gentlemen..."
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
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.
Interestingly solved! Great video. My favorite ones are 2 and 5. Thanks a lot.
Fantastic video: It definitely makes me study much harder than I used to-
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
Idk why I’m watching this as I’m in precalc but this is some dark magic holy shit.
Amazing! The last Integral was cool.
Woah this very well amazing. Thanks for MIT Integration this year and solving all problems. 🥰🥰😝
How did you manage to break me before 3 seconds of videos with just "ladies and gentlemen" LMAO, simply absolute cinema.
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
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
I might have used Euler substitution at the sqrt(x²±x+1). It happens so rarely but is so satisfying
Summation inside of an integral
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)
Timestamps
Q1 : 0:11
Q2 : 4:49
Q3 : 12:26
Q4 : 21:24
Q5 : 25:05
The ability to integrate beyond target, is what separates an obtuse predator from an acute producer.
The first good math video on youtube ❤❤
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.
Only bee that I integrated with was a bee that stung me
😂😂😂
kudos for the efforts!! i hve been finding the solutions to these probs but noone posted a video or something. Thanks
For the first problem you can simply sub z = u^1/3 and then solve it using elementary techniques!
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.
21:22
It is actually ln(2+sqrt(7)/sqrt(3)) at the end, not just 3
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?
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.
Thank you so much for this video ❤
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.
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. 😊
I would’ve done residue theorem for the second one but I never pass up an opportunity for that
C’est vraiment incroyable que ils peuvent resoudre celles problèmes aux quatre minutes.
at 15:05 why didnt you take a - b = +x and a- b = -x as two different cases?
Maybe we imply that x>0
25:02 Are you simply going to skip the floor function part of problem 4? The whole challenge is to find an integer solution to the floor of the answer.
At 12:02 Small mistake that does not affect the result, but you forgot a factor of 1/2 from the product to sum identity.
In the end, the result is still 0.
You could’ve avoided a lot of the mess in problem 5 by simply cancelling the 2^(n-1) and the 2^n at 36:09, leaving 1/2, instead of expanding and inducing errors.
bro is the integral paladin
🤓
the last integral is something of its own😵💫did you have the same reaction while solving🤣:D
I definitely enjoyed it
This integration from Beta and gamma function and good question to explain all how to gama function work 4:52
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?
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))
Thank you sooooo much. Really appreciate
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!
Guys I made it to 1:36 yay. No further tho
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
1. I'm not as smart as the bois at MIT
2. They don't have to explain anything while solving😂
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
@@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
@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.
crazy how the mit kids do it in like 4 minutes
That really was a wild ride
This is beautiful and artistic
I'm pretty proud that you only lost me at the beta and gamma functions
What should I know about integrals for do this kind of integrals.
Literally me during the entire video 8:28
14:13 2x^2
1:08
This is very much not clear as day and iam very confused.
Im not even sure how i got here.
So are you not going to do the fourth problem with the floor function? The floor function is what actually makes it difficult
Why would we skip the floor function bit?
That's the whole point of the integral!
@@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
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.
@@maths_505Yes, but how do you evaluate the floor of the answer for that problem?
There should be a side contest for people integrating with approximations like Riemann Sums and Simpson’s Method.
Honestly problem 6 was done by me but in 15 minutes
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)
You can solve JEE PYQs.
This is more complicated than all the problems and issues that I have to deal with.
th-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
"pretty much clear as day" >.>
Loved the first one... going to watch the rest... ❤❤
13:10 Bro u have any idea where i could learn such helpful algebra theories? Any names for these concepts?
Binomial square?
@@yodaas7902 thanks
@@yodaas7902 thanks
I am here just to say I saw the thumbnail and immediately started crying from PTSD.
Sorry about that mate😂
Damn!! I was able to answer some of them.
I had already solved 3 of them before seeing this video, still 2 left to solve......
Il primo è semplice B(4/3,2/3)
WHAT KIND OF WAY IS THAT TO WRITE π
+ x y 3/tanx 3+3+3=9 + 3X3=9 x/x=1 1-3/ / pi/2 2 pi /3/9
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
What is beta and gamma function first time heard about it. I never heard about it earlier
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
The solution to the first problem demonstrates why I failed again and again via partial fraction stuff...
So many times in this video, I pause for a few minutes and then say oh I'm stupid, then continue watching
Kinda skipped some steps at the start of 2nd problem 4:49 lol
Oh damn
Thanks for pointing that out mate
@@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
@@par22 thanks mate
For the record going from step 1 to step 2 uses the double angle formula for the sine function
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 😐.
You'll find lots of tough integrals in higher physics and applied math....these are just like practice problems to help think
@@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.
@@alien77333 😂😂😂😂😂😂😂
En el tercer ejercicio te faltó poner al tres dentro de la raíz cuadrada en el resultado final de la integral.
FYI, problem 4 had the whole integral under a FLOOR FUNCTION.
I think in 14:03 it should be 2x^2 and not 2x
lmao, i should have watch first before leaving this comment. haha. 14:35 it is
I watched this right after taking Calc AB. Im scared 😂
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
11th grader me watching it and having absolutely no idea about whatever is going on😂😂😂
/pi 0 ( cosx/cos 3x cos 15x/cos5x)2 dx
I wish you were my best friend all throughout my math major.
You forgot to timestamp Q5. It's around 24:13
EDIT: Not sure what I was thinking when I wrote that. It's 25:03
how the heck do they come up with these integrals that end up in simple solutions?
3:27 As a russian, I understand. Sorry about that 😓
Мне кажется, что здесь это ни при чëм 🙂
@@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
Which program do you used to write on the phone? (i mean the black board)
I want a professor like you to make math fun again in university
Hello fellow Yu-Gi-Oh fan
@@yuseifudo6075 I see you are a man of culture as well
Naaaa they had to write out the digits in integral 4. There was also a floor function around the whole thing, hence the 10^20.
He is the Formula King!!!