thanks , the integral is really amazing. so intelligent. Mathematics is deeply bottomless, and probability and statistics are more difficult to understand. Gamma distribution, chi-square distribution, is so hard to understand.
This is probably the most fun I've ever had watching a video about integrals! It was a long journey, but it was worth every second. Thankyou for such a great video!
@@blackpenredpen But why on earth would you do that second substitution at all? Isn't there another way maybe more intuitive..like integration by parts Ibwas thinking..Hope you can respond when you can.
Evolution can you give me more letters for my substitions into different integral worlds? Sure blackpenredpen ACTUALLY USES OLD LETTERS FOR SUBSTITUTIONS LIKE A BOSS
when you say "please do not ask me to check the answer by diferentiation", I'm pretty sure you were really thinking "please ask me to check the answer by diferentiation" ... So, in order to please you..... Would you check the answer by diferentiation???? Please :)
scrolled down expecting the comment "check the answer by differentiation" as top or only comment .. urs was 3rd and has a bunch of text around it internet, you disappoint...... (altho the expectation ur not meeting, is actually... so in a way.... yay! ....)
16:56 Yeah, putting down some TNT on that would definitely help :) 18:18 It's quite interesting that `1/2` and `√3/2` appear in the completed square, because: a) they are the real and imaginary part of the cube root of `-1` that appear in the complex factorization of the denominator; b) they are the sine and cosine of the 60° angle at which this cube root lays with respect to the real axis :) c) we've got the cube root (of the tangent) in the original integrand. 21:30 We have to go deeper... :> 23:52 The dream is collapsing :J
I was wondering this myself lol at the beginning of this video. I'm betting it inevitably leads to just letting u= the n-th root of tan(x) and then integrating the resulting rational function using partial fraction decomposition - maybe synthetic division for powers of tangent not = 1 or 2.
This should be easy. Separate tan^n into tan^(n-2) and tan^2 then integrate by parts. Then substitute sec^2-1 for tan^2 and you get a reduction formula. Evaluate integral of tan^2 and that should be enough I guess.(I havent tried this but I think this works for even exponents only.)
The second integral can be done without any substitution: Multiply and divide by 2 (you can factor out the divide by 2 later, with the final constant distribution): (2t+2)/(t²-t+1) The derivative of the denominator is 2t-1. So, write the integral as: (2t-1+3)/(t²-t+1) Split the fraction: (2t-1)/(t²-t+1) + 3/(t²-t+1) The first one's integral is ln|t²-t+1|. For the second one, complete the square as (t-1/2)²+(√3/2)². The integral is 2√3*invtan((2t-1)/√3). Now, distribute the divide by 2: 1/2*ln|t²-t+1|+√3*invtan((2t-1)/√3).
This couldn't have been uploaded at any time better, I'm just about to fall asleep and I always trance at your videos which makes me go to sleep! Thank you so much! :D
I thought he will forget the + C part Truly professional to the core I am starting to fall in love with math and pens And of course our one and only *CALCULUS*🥳🥳
i never´ ve seen a intgral like this . i been watching all your videos and in my opinion is very useful. in other side i want to thank, you videos are helping me to improve my english how you know ,y native language is spanish, but thanks for share it. congrats
i just love your videos man... i just want to say that you helped me overcome my fear of integrals.. and your approach to not just this video but all the others are just brilliant.. thank you.
En 6:10. Dentro de las integrales NO se deben mezclar las variables. Por eso se llama CAMBIO DE VARIABLES. En integrales MULTIPLES sí pueden haber mas de una variable
This channel is now offically a MATH MEME ! "How about the integral of [fancy variation of sqrt(tan x)] ?" - bprp, you have a long meme carrier ahead...
15:20 i think we can multiply 2 then having a division before completing the square 2(t+1)-3+3/t*2-t+1=2t-1/t*2-t+1 +3/t*2-t+1,so that we do not need to do substitution with w
I kept on making mistakes when trying to check this answer by differentiation, and had pledged to keep this video tab open until I had successfully checked it. So pleased to say that I finally got it right today! I can close the tab now. :D
I want to see this too, although I'll try it myself first because it may be possible that it be not expressible in terms of elementary functions or with a finite definite expression.
The interesting thing would be a professor taking the solution of that integral and making the class find the derivative maybe as a bonus problem on an exam.
So how does the monster grow in the complex domain? I would love to see the secrets in there so that I might apply them to quantum mechanics (and my cool design for a new warp drive model).
I know this is 5 years later but dont forget to divide the lns. Also for partial fractions you can multiply the denominators on the left side to make it easier. Hope this helps
if they´re not nice they get this integral on their exam! how about that? or worse sin(x)/x from 0 to infinity (but don´t ask for the steps. that´s overkill. though maybe you can show us that one? i didn´t see a single video on YT about that ntegral i would have understood. and the only idea to even make this possible would be using the mclaurin series of the sine)
Standard integrals are square root of tangent x, tangent x elevated to 1/3 and so on like tangent x elevated to 1/4. The exercise gets extremely complicated when we consider powers of 1/5 and so. I think that tangent to the 1/2 and tangent to the 1/3 are fair practice problem for assignments covering almost every technique in integration. This is a good exercise for substitution, partial fraction, determinants and the Gauss Jordan elimination method. It has the classical strategies and the knowledge of well known integrals including the logarithmic ones.
...And when you want to take a break from listening to this video, you are going to SUBSTITUTE IT for one of the easier/more beautiful ones to understand :D Like sin(z)=2
"and then we have to get back to the U world, and then we have to go back to the X world." What a wild ride - we're going on interplanetary journeys solving this integral!
Just substitute the i's in the right places! tan(x) = -i tanh (ix), so you get tanh(x)=i tanh(-ix). Substitute this into the original integrand you get the cube root of i which is exp(i*pi/6), then you get an i outside when you replace dx with d(-ix). So the answer to integrating cube root of tanhx is just the original answer with all the x's replaced by -ix's, and an extra factor of exp(i*pi*2/3). Somehow you should be able to simplify everything in the end expression to show that the thing is real XD
I think its so strange that this integral is so tough, but i could solve it myself, like i understood every step, i knew what to do to get there myself, it just takes so many steps. Thats the first time ive been able to say that about a tough maths problem on youtube, its a good feeling.
OK, I checked the answer and it was correct! th-cam.com/video/mUXdgY5MeEg/w-d-xo.html
Happy New Year!!!
I approve.
Lol, just finished the vid and relaised this
Oh god my maths test is on 1st Aug and I haven't prepared for it yet...... integrals is also in test!😢😢😢
thanks , the integral is really amazing. so intelligent.
Mathematics is deeply bottomless, and probability and statistics are more difficult to understand.
Gamma distribution, chi-square distribution, is so hard to understand.
i really need your help to intigrate this for me i really need it ----- x-1*sqrt.2x-x*2
/x+1
Somewhere out there, a professor is gonna watch this video and make this problem a test question. RIP.
Sheev Palpatine Saw the comment, liked for the name
Sheev Palpatine I just had to pin this comment.
oh no, don't give them ideas
Sheev Palpatine that would be acceptable if that was an extra credit question where if you get it right you get full credit for the exam
Sheev Palpatine did you ever hear the tragedy of darth plagueis the wise?
You know it's a hard problem when he has to use 3 colors
BT7M yup
blackpenredpenbluepen
This is not hard prblm we indians take 30 sec to solve this
@@g.peloni3694 none cause we never and then we're like wtf are we doing.
@@g.peloni3694 uff...😂I'm also An Indian I'm a Calculus Lover too...
Check the answer by differentiation
You mad man
You'll break him
I didn't understand the joke until I saw the answer XD
Avi Mehra The mad lad actually did it
Integrate it again.
8:15 *erases that additional line*
ohh god thank youu
yoav carmel lollllll nice catch!!!!
Android Ninja yup
Wow
This is probably the most fun I've ever had watching a video about integrals! It was a long journey, but it was worth every second. Thankyou for such a great video!
Riley Wells my pleasure!!!!
@@blackpenredpen But why on earth would you do that second substitution at all? Isn't there another way maybe more intuitive..like integration by parts Ibwas thinking..Hope you can respond when you can.
I don't want you to check your answer by differentiation. I want you to integrate your answer :D
BRUH
xD
You're a monster
That’s easy
@@asparkdeity8717 yeah ok
I think you did a step wrong somewhere, please differentiate it to check.
Lolololol
pls no 😭
You're going from blackpenredpen to 3blue1brown
How so? His vids are way cooler
Because this is really REALLY complicated yet you manage to explain it so well. Its cool at the same time
blackpenredpen Because of the color markers you are using
blackpenredpen and sometimes blue : )
blackpenredpenbluepengreenpen
I love this. Great example of when to actually use partial fractions, too. I'm adding this channel to my list of favorite math channels! 🤓🤓🤓
Natalie Euley thank you Natalie!
I fear you may run out of substitute variables one day :P
Shakir Ahsan Romeo probably. Next I will use alpha, beta
*creates new alphabet after using all latin, greek, cyrillic, hebrew and arabic letters*
Evolution can you give me more letters for my substitions into different integral worlds?
Sure blackpenredpen
ACTUALLY USES OLD LETTERS FOR SUBSTITUTIONS LIKE A BOSS
After that start using ♤♡◇♧¤▪☆
How about हिन्दी
when you say "please do not ask me to check the answer by diferentiation", I'm pretty sure you were really thinking "please ask me to check the answer by diferentiation" ... So, in order to please you..... Would you check the answer by diferentiation???? Please :)
lol! I am waiting for some viewers to do it!
blackpenredpen We want you to check it by differentiation. It's always easier than the integration, right?
blackpenredpen I proved it using my calculator. Is that good enough?
scrolled down expecting the comment "check the answer by differentiation" as top or only comment ..
urs was 3rd and has a bunch of text around it
internet, you disappoint......
(altho the expectation ur not meeting, is actually... so in a way.... yay! ....)
Maybe?!
16:56 Yeah, putting down some TNT on that would definitely help :)
18:18 It's quite interesting that `1/2` and `√3/2` appear in the completed square, because:
a) they are the real and imaginary part of the cube root of `-1` that appear in the complex factorization of the denominator;
b) they are the sine and cosine of the 60° angle at which this cube root lays with respect to the real axis :)
c) we've got the cube root (of the tangent) in the original integrand.
21:30 We have to go deeper... :>
23:52 The dream is collapsing :J
Ah!!! I like the pt u made at 18:18... and what u said "The dream is collapsing"
30:40 "Please, do not ask me to check the answer by differentiation"
Hey, you should totally check the answer by differentiation!
Worth every second of the watch. Thank you!
200%productions thank you!!
I have a challenge for you. Make the generalization of the integral of n-th root of tan x dx
I was wondering this myself lol at the beginning of this video. I'm betting it inevitably leads to just letting u= the n-th root of tan(x) and then integrating the resulting rational function using partial fraction decomposition - maybe synthetic division for powers of tangent not = 1 or 2.
of course, I'm ignoring any possible trig manipulations ...
This should be easy. Separate tan^n into tan^(n-2) and tan^2 then integrate by parts. Then substitute sec^2-1 for tan^2 and you get a reduction formula. Evaluate integral of tan^2 and that should be enough I guess.(I havent tried this but I think this works for even exponents only.)
Kyro well Oh yeah..
Lol that's sadistic!
Doctor: You have 31 minutes and eight seconds to live.
Me: immediately opens this video
Now that is just incredible! By far one of the most difficult integrals I have ever seen and you have solved it so elegantly and so well.
The second integral can be done without any substitution:
Multiply and divide by 2 (you can factor out the divide by 2 later, with the final constant distribution):
(2t+2)/(t²-t+1)
The derivative of the denominator is 2t-1. So, write the integral as:
(2t-1+3)/(t²-t+1)
Split the fraction:
(2t-1)/(t²-t+1) + 3/(t²-t+1)
The first one's integral is ln|t²-t+1|. For the second one, complete the square as (t-1/2)²+(√3/2)². The integral is 2√3*invtan((2t-1)/√3).
Now, distribute the divide by 2:
1/2*ln|t²-t+1|+√3*invtan((2t-1)/√3).
Video: “Easiest integral on TH-cam”
Me: oh that’s nice let’s check it out
Also video: equation is written very small and 31 minutes
Me: oh no...
This couldn't have been uploaded at any time better, I'm just about to fall asleep and I always trance at your videos which makes me go to sleep! Thank you so much! :D
This problem was soooooo easy! I solved it when I was a foetus. Hats off to this young gentleman realising the foetal power of solving integrals!
Fetus?
Bro really spelled it with an o 💀
@@JMZReview diarrhoea
30:40 differentiASIAN
Lmao
Now do the integral of (tanx+(tanx)^(1/2)+(tanx)^(1/3)+(tanx)^(1/4)+(tanx)^(1/5)...)
Mr. Gentlezombie spoiler alert
this series will diverge, this question doesn't make sense.
@@aneeshsrinivas9088 D'oh!
Ok to the one dislike on the video I want you to post a follow up showing everyone how you do this integral.
Mace Jr there is no dislike now
NestorV S Hhh he couldn't do it.
Nah two people did.
disliking =/= "i can do better"
ask any sports fan if you don't believe me.
Now 5³ dislikes
For B in 11:45, I like to multiplie by t in both side, and by going in the limit in +infini, we have 0=A+B
I thought he will forget the + C part
Truly professional to the core
I am starting to fall in love with math and pens
And of course our one and only
*CALCULUS*🥳🥳
I love how enthusiastic you are! What great videos you make.
Fantastic job of explaining every step!
what about the tan(x) root of tan(x)?
I think impossible
NestorV S I thought you could integrate anything you wanted
Ace Eternus no, there are functions that their integral have no solution. A classic example of this is the function f(x)=e^x^2.
Ace Eternus
Some functions have no elementary anti derivative take x^x for example, you have to settle for a series.
Gregory House Tecnically there is a solution, but it just can't be expressed in terms of exponentials, polynomials, logarithms etc
Imagine doing a test and solving this problem. You did every step right, but you forgot to write plus C
And professor deducts 1mark for that
@@delta2884 my professor would tear the whole thing and give you 0 ;)
Easier when in France they ask for a antiderivative so you don't have to add +C
Calc II + Calc II
Best problem ever!
Tour de force for methods of integration.
I counted eight different tricks.
TI-84 plus showed it correct over:
-2π
Can you check the answer by differentiation ?
Pedro Bessa maybe one day :)
Really quick way to check. Takes 30 seconds. Plug in some values the the limit of integration and use fund. Thm of calc p2.
Could you check the answer by differentiation?
no
yes
Maybe
i never´ ve seen a intgral like this
. i been watching all your videos and in my opinion is very useful. in other side i want to thank, you videos are helping me to improve my english how you know ,y native language is spanish, but thanks for share it. congrats
Thank you so much. I am very happy to hear your nice comment! Keep up the good work too!
i just love your videos man... i just want to say that you helped me overcome my fear of integrals.. and your approach to not just this video but all the others are just brilliant.. thank you.
Please differentiate this to get back to cuberoot of tan x
En 6:10. Dentro de las integrales NO se deben mezclar las variables.
Por eso se llama CAMBIO DE VARIABLES.
En integrales MULTIPLES sí pueden haber mas de una variable
This channel is now offically a MATH MEME !
"How about the integral of [fancy variation of sqrt(tan x)] ?" - bprp, you have a long meme carrier ahead...
oscarjd74 OH WOW!!!! YOU ACTUALLY DID IT!
NEAT, instant screen saver ^^
How cool iz dat?! HAHAHAH
Watching this made me very relaxed. You sir deserve a subscribed!
here i am watching 30 minutes of math porn and not knowing wtf is going on
Clement Tan isn’t that the case when watching regular porn too?
Жиза
25:49 let's all appreciate how his only mistake in the video was using the wrong coloured marker
please check the answer by differentiation
15:20 i think we can multiply 2 then having a division before completing the square 2(t+1)-3+3/t*2-t+1=2t-1/t*2-t+1 +3/t*2-t+1,so that we do not need to do substitution with w
I kept on making mistakes when trying to check this answer by differentiation, and had pledged to keep this video tab open until I had successfully checked it. So pleased to say that I finally got it right today! I can close the tab now. :D
BlackFiresong wow nice!!! I will check it one day too.
Thank you so much for replying! Love your work. :) Please do keep the crazy challenges coming. You have really helped to rekindle my love of Maths!
BlackFiresong I am very glad as well!! Thank you for watching my videos :)
Glad you enjoyed yourself with that. This is a great review of a lot of techniques.
He did the derivative!!!!!!!!!!!! Check out Fematika's
th-cam.com/video/as1vk0IuAHA/w-d-xo.html
you are such an amazing teacher you make this advanced stuff seem so easy
Please make a video about the nth root of tanx
yes i agree
Yes, that'd be great! :)
I want to see that too
tan(nx) ^ (1/n)
I want to see this too, although I'll try it myself first because it may be possible that it be not expressible in terms of elementary functions or with a finite definite expression.
Gratulálok, ez a szinte végig helyettesítéses módszer bemutatta, hol kezdődik az elméleti matematika !
Great, now integrate (tanx)^1/5
The interesting thing would be a professor taking the solution of that integral and making the class find the derivative maybe as a bonus problem on an exam.
This does turn out to be a monster of an integral "doesn't it" :-)
Yea, isn't it?!
So how does the monster grow in the complex domain?
I would love to see the secrets in there so that I might apply them to quantum mechanics (and my cool design for a new warp drive model).
Jesus Christ! With this you have won a new follower! Subscribed!
I was kind of hoping to see you dancing in celebration chanting "One Take!" at the end
who knows maybe its 10th take
Notice that I didn't edit this video!
MrQuantum "one take" and "1st take" are different.
Mark Zero true. One take for this tho. I think it was my 3rd take bc my intros weren't good enough.
Mark Zero k
I know this is 5 years later but dont forget to divide the lns. Also for partial fractions you can multiply the denominators on the left side to make it easier. Hope this helps
Can you make some videos about the strategies of choosing the right integration technique for different problems?
Bon Bon will try.
very very good video and very satisfying to have a clean solution in the end.
You earned yourself a new sub
Thanks sir .
Glad to hear! Thank you!
I'm so happy to see a "BIG INTEGRALS NO SHORTCUTS" playlist, will watch them all
I'm worried for you.
MistaTwoJeffreyTwenty Yaay
U should worry about my students
blackpenredpen oh noooo, show them sympathy please, they don't deserve this!
Lol! I am teaching a calc2 class in the fall. We will see!! : )
blackpenredpen the next year i will see integrals on our tests and i need a bless from Pitagoras to pass these exams
if they´re not nice they get this integral on their exam! how about that? or worse sin(x)/x from 0 to infinity (but don´t ask for the steps. that´s overkill. though maybe you can show us that one? i didn´t see a single video on YT about that ntegral i would have understood. and the only idea to even make this possible would be using the mclaurin series of the sine)
Most elegant integral I've seen so far,thank you sir
Nothing even remotely elegant about this. This is the equivalent of opening a safe by dropping an anvil on it.
This is so great! I wish I still had the "mathematical muscle" to do this myself. Maybe one of these days...
Hey man just checking in, its been 4 years hows that math muscle working out for you? I challenge you to do 3 integrals! You can do it
@@thelegendofme7520 Aaany day now... ;)
"The Easiest Integral on TH-cam" = kalm
*Sees the video length **31:08**" = PANIC*
: )))))
Standard integrals are square root of tangent x, tangent x elevated to 1/3 and so on like tangent x elevated to 1/4. The exercise gets extremely complicated when we consider powers of 1/5 and so. I think that tangent to the 1/2 and tangent to the 1/3 are fair practice problem for assignments covering almost every technique in integration.
This is a good exercise for substitution, partial fraction, determinants and the Gauss Jordan elimination method. It has the classical strategies and the knowledge of well known integrals including the logarithmic ones.
Good. this is a high level problem in my sheet .
You're a braver man than I. Damn that's one intense integral.
do the integral of sqrt(tan(cuberoot(tanx)))
Kyle Farias Just use numerical integration on given bounds
And my professor is expecting me to solve that in only 10 minutes.
"When in doubt, use horseshoe mathematics."
You too saw the video!!
I see, you are a man of "NO U" culture here!
At 18:00 , why not just use trig substitution. I tried it using 3^(1/2)/2 tan theta. It worked out alright.
pure beauty...!
Thank you!!!!
Easiest integral on TH-cam
31 min yt video - "no i don't think i will"
When title says easy and time says half an hour
🤔🤔🤔🤔🤔🤔🤔😱
Cringe
This is just what I needed for my summer.
yea dog imma need you to check by differentiation
It's also expressible in terms of "the" hypergeometric function 2F1.
Can you do integral of (xtanxdx) ?
The exasperation at 25:00, "This is not the whole question!!" XD
And again at 29:45 "We're almost done, oh my god!"
Harys Page lollll
Running a marathon is easy compared to solving this integral 🤓
You're videos are awesome! Please keep posting :)
It kinda bothers me that you write "tan^-1" instead of "arctan", it makes it seem like you could get tan^-(1/3) by multiplying it with tan^(2/3).
Florian Bender if you mean 1/tan x, he would write it as cot x
Florian Bender tan^-1 is a standard notation for arctan
I agree. It's bad form IMO to both use tan^-1(x) for arctan(x) as well as the tan^q(x) for (tan(x))^q within a single exercise.
Monzur Rahman We're aware, but it's a terrible notational standard.
It's not even a "standard" notation really. Just a commonly used one (and not even as commonly used as the arctan notation).
To solve for B and C in partial fraction decomposition do the same as u did for A but find for that expression a complex root, substitute then compare
Okay so you've done sqrt tan x, you've done cube root tan x, can you generalize? Nth root tan x ?
I love how you are so passionate about maths
thank you!!!
Dr. BalckPenRedPen, please check the answer by differentiation :)
seems like a deep breathtaking moment when you got the answer to be this long...
...And when you want to take a break from listening to this video, you are going to SUBSTITUTE IT for one of the easier/more beautiful ones to understand :D Like sin(z)=2
"and then we have to get back to the U world, and then we have to go back to the X world."
What a wild ride - we're going on interplanetary journeys solving this integral!
It is too long, not too hard. Anyway I doubt anyone can do it in any exam without making a small mistakes ☺
it's like running a marathon (well, almost...)
You know what you have to do now? Now you have to do the integral of (tanhx)^(1/3). These videos are the best please keep posting them!
Tom Himler Edward can help me with that. :)
Just substitute the i's in the right places! tan(x) = -i tanh (ix), so you get tanh(x)=i tanh(-ix). Substitute this into the original integrand you get the cube root of i which is exp(i*pi/6), then you get an i outside when you replace dx with d(-ix). So the answer to integrating cube root of tanhx is just the original answer with all the x's replaced by -ix's, and an extra factor of exp(i*pi*2/3). Somehow you should be able to simplify everything in the end expression to show that the thing is real XD
now the nth root
If every question on the test is like this Mr. Chow, I might have to clear my whole weekend to complete it.
Welcome to Calc2!
: )
....It wasn't that bad....it didn't have limits to substitute (hehe ! )
Integrand has a period of pi. It's real-valued from [0,pi/2) and complex-valued from (pi/2,pi). Goes to +inf at pi/2.
To be fair, I have dealt with much tougher problems. Even on TH-cam.
share, yea?
*smashes bottle on counter, holding its jagged remains menacingly* yea?
Wow you so smart you must watch Rick and Morty
Rick and Morty is a fun show, nothing to do with being smart lol
I didn't mean to say this is an easiest integral lol. It is a tough problem, but not the hardest I meant to say
I think its so strange that this integral is so tough, but i could solve it myself, like i understood every step, i knew what to do to get there myself, it just takes so many steps. Thats the first time ive been able to say that about a tough maths problem on youtube, its a good feeling.
dude same 😭
To think i can finally keep up with some videos on this channel
Everytime he says 'something' I hear 'sensei' *_*
Samarth Varshney or... passdee instead of positive. Forgive him, neither him or me are english ppl
That was a really enjoyable lil’ video. Thank you, it was a nice little refresher! 🙂