The Easiest Integral on YouTube

He did the derivative!!! Check out Fematika • Checking BlackPenRedPe...
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blackpenredpen

[@blackpenredpen]

OK, I checked the answer and it was correct! kzbin.info/www/bejne/o4a7lZqParKYe8k Happy New Year!!!

[@notnilc2107]

I approve.

[@alexsanders7303]

Lol, just finished the vid and relaised this

[@marziyaali7962]

Oh god my maths test is on 1st Aug and I haven't prepared for it yet...... integrals is also in test!😢😢😢

[@sahlileila6382]

i really need your help to intigrate this for me i really need it ----- x-1*sqrt.2x-x*2 /x+1

[@Jared7873]

KZbin viewers say this is the easiest integral (31 minutes to solve). 😑

[@wellbangok8959]

Somewhere out there, a professor is gonna watch this video and make this problem a test question. RIP.

[@davidkippy101]

Sheev Palpatine Saw the comment, liked for the name

[@blackpenredpen]

Sheev Palpatine I just had to pin this comment.

[@levi4328]

oh no, don't give them ideas

[@Mathemusician97]

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

[@神-w1y]

Sheev Palpatine did you ever hear the tragedy of darth plagueis the wise?

[@BT7M]

You know it's a hard problem when he has to use 3 colors

[@blackpenredpen]

BT7M yup

[@isaacormesher2851]

blackpenredpenbluepen

[@Anonymous_Poetry]

This is not hard prblm we indians take 30 sec to solve this

[@booga9584]

@@g.peloni3694 none cause we never and then we're like wtf are we doing.

[@midhun.m4963]

@@g.peloni3694 uff...😂I'm also An Indian I'm a Calculus Lover too...

[@AviMehra]

Check the answer by differentiation

[@jeffreycanfield1939]

You mad man

[@S-Matrix]

You'll break him

[@SteamPunkLV]

I didn't understand the joke until I saw the answer XD

[@federicovolpe3389]

Avi Mehra The mad lad actually did it

[@nellvincervantes3223]

Integrate it again.

[@friendofjim]

I don't want you to check your answer by differentiation. I want you to integrate your answer :D

[@dhruvkaushik4872]

BRUH

[@StoryTime-jo6ij]

xD

[@bananoramatfw]

You're a monster

[@asparkdeity8717]

That’s easy

[@random22453]

​@@asparkdeity8717 yeah ok

[@yoavcarmel1245]

8:15 *erases that additional line* ohh god thank youu

[@blackpenredpen]

yoav carmel lollllll nice catch!!!!

[@blackpenredpen]

Android Ninja yup

[@yrcmurthy8323]

Wow

[@flansterr]

I think you did a step wrong somewhere, please differentiate it to check.

[@Vibranium375]

Lolololol

[@abhirupkundu2778]

pls no 😭

[@davidkippy101]

You're going from blackpenredpen to 3blue1brown

[@blackpenredpen]

How so? His vids are way cooler

[@Number3pt14]

Because this is really REALLY complicated yet you manage to explain it so well. Its cool at the same time

[@davidkippy101]

blackpenredpen Because of the color markers you are using

[@blackpenredpen]

blackpenredpen and sometimes blue : )

[@NoNameAtAll2]

blackpenredpenbluepengreenpen

[@rileywells3045]

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]

Riley Wells my pleasure!!!!

[@leif1075]

@@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.

[@shakirahsanromeo7863]

I fear you may run out of substitute variables one day :P

[@blackpenredpen]

Shakir Ahsan Romeo probably. Next I will use alpha, beta

[@iconoptixx]

*creates new alphabet after using all latin, greek, cyrillic, hebrew and arabic letters*

[@Queenhideyxo]

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

[@arnavanand8037]

After that start using ♤♡◇♧¤▪☆

[@hamiltonianpathondodecahed5236]

How about हिन्दी

[@andresvivas2814]

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 :)

[@blackpenredpen]

lol! I am waiting for some viewers to do it!

[@manuelbonet]

blackpenredpen We want you to check it by differentiation. It's always easier than the integration, right?

[@monzur1947]

blackpenredpen I proved it using my calculator. Is that good enough?

[@aysikl]

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! ....)

[@blackpenredpen]

Maybe?!

[@tipoima]

30:40 "Please, do not ask me to check the answer by differentiation" Hey, you should totally check the answer by differentiation!

[@maxhaibara8828]

I have a challenge for you. Make the generalization of the integral of n-th root of tan x dx

[@kamarinelson]

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.

[@kamarinelson]

of course, I'm ignoring any possible trig manipulations ...

[@eroskuikel9688]

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.)

[@eroskuikel9688]

Kyro well Oh yeah..

[@surenmoodley7744]

Lol that's sadistic!

[@natalieeuley1734]

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! 🤓🤓🤓

[@blackpenredpen]

Natalie Euley thank you Natalie!

[@bonbonpony]

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

[@blackpenredpen]

Ah!!! I like the pt u made at 18:18... and what u said "The dream is collapsing"

[@JoJoJet100]

Doctor: You have 31 minutes and eight seconds to live. Me: immediately opens this video

[@Kreighzey]

30:40 differentiASIAN

[@particleonazock2246]

Lmao

[@mr.gentlezombie8709]

Now do the integral of (tanx+(tanx)^(1/2)+(tanx)^(1/3)+(tanx)^(1/4)+(tanx)^(1/5)...)

[@hauler-hx6gw]

Mr. Gentlezombie spoiler alert

[@aneeshsrinivas9088]

this series will diverge, this question doesn't make sense.

[@mr.gentlezombie8709]

@@aneeshsrinivas9088 D'oh!

[@ianbo1]

Worth every second of the watch. Thank you!

[@blackpenredpen]

200%productions thank you!!

[@azmath2059]

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.

[@malayagr]

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).

[@blackpenredpen]

He did the derivative!!!!!!!!!!!! Check out Fematika's kzbin.info/www/bejne/l6SUp55mftp0fqM

[@Xperia347]

Could you check the answer by differentiation?

[@moregirl4585]

no

[@awawpogi3036]

yes

[@banana6108]

Maybe

[@particleonazock2246]

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!

[@lih3391]

Fetus?

[@JMZReview]

Bro really spelled it with an o 💀

[@General12th]

@@JMZReview diarrhoea

[@Onlytopup6969]

😂😂

[@12345678910111213796]

Can you check the answer by differentiation ?

[@blackpenredpen]

Pedro Bessa maybe one day :)

[@nationalstudyacademykim5030]

Really quick way to check. Takes 30 seconds. Plug in some values the the limit of integration and use fund. Thm of calc p2.

[@loganmancuso3791]

Ok to the one dislike on the video I want you to post a follow up showing everyone how you do this integral.

[@nestorv7627]

Mace Jr there is no dislike now

[@aceeternus4027]

NestorV S Hhh he couldn't do it.

[@ZyloSol99]

Nah two people did.

[@busTedOaS]

disliking =/= "i can do better" ask any sports fan if you don't believe me.

[@hamiltonianpathondodecahed5236]

Now 5³ dislikes

[@X-3K]

what about the tan(x) root of tan(x)?

[@nestorv7627]

I think impossible

[@aceeternus4027]

NestorV S I thought you could integrate anything you wanted

[@RubALamp]

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.

[@mh-no5it]

Ace Eternus Some functions have no elementary anti derivative take x^x for example, you have to settle for a series.

[@sinistrolerta]

Gregory House Tecnically there is a solution, but it just can't be expressed in terms of exponentials, polynomials, logarithms etc

[@ATeamBrown]

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

[@sethleverett8889]

Video: “Easiest integral on KZbin” Me: oh that’s nice let’s check it out Also video: equation is written very small and 31 minutes Me: oh no...

[@rohitg1529]

Please differentiate this to get back to cuberoot of tan x

[@mathematically4761]

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*🥳🥳

[@mutenfuyael3461]

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

[@stanschimerman]

Fantastic job of explaining every step!

[@redvine1105]

You're a braver man than I. Damn that's one intense integral.

[@eveliohernandez6595]

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

[@blackpenredpen]

Thank you so much. I am very happy to hear your nice comment! Keep up the good work too!

[@howtoglobal7901]

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.

[@DreamzAnimation]

please check the answer by differentiation

[@dovidglass5445]

I love how enthusiastic you are! What great videos you make.

[@void2509]

Imagine doing a test and solving this problem. You did every step right, but you forgot to write plus C

[@delta2884]

And professor deducts 1mark for that

[@invasion8318]

@@delta2884 my professor would tear the whole thing and give you 0 ;)

[@dzlcrd9519]

Easier when in France they ask for a antiderivative so you don't have to add +C

[@好吧-h6k]

Calc II + Calc II

[@君爵_2045-y7k]

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

[@tavishu]

Great, now integrate (tanx)^1/5

[@andrecouverette]

Best problem ever! Tour de force for methods of integration. I counted eight different tricks. TI-84 plus showed it correct over: -2π

[@bonbonpony]

Can you make some videos about the strategies of choosing the right integration technique for different problems?

[@blackpenredpen]

Bon Bon will try.

[@joserubenalcarazmorinigo9540]

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

[@philmertens2673]

Please make a video about the nth root of tanx

[@TheTimeDilater]

yes i agree

[@dicandeo9698]

Yes, that'd be great! :)

[@sunriselp849]

I want to see that too

[@jiaming5269]

tan(nx) ^ (1/n)

[@angelmendez-rivera351]

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.

[@guydht1]

30:33 Can you *Please* check your answer by differentiation?

[@BlackFiresong]

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

[@blackpenredpen]

BlackFiresong wow nice!!! I will check it one day too.

[@BlackFiresong]

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!

[@blackpenredpen]

BlackFiresong I am very glad as well!! Thank you for watching my videos :)

[@kummer45]

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.

[@kylefarias9143]

do the integral of sqrt(tan(cuberoot(tanx)))

[@nejlaakyuz4025]

Kyle Farias Just use numerical integration on given bounds

[@MRBYRNO2505]

you are such an amazing teacher you make this advanced stuff seem so easy

[@jeffreycanfield1939]

I'm worried for you.

[@blackpenredpen]

MistaTwoJeffreyTwenty Yaay U should worry about my students

[@jeffreycanfield1939]

blackpenredpen oh noooo, show them sympathy please, they don't deserve this!

[@blackpenredpen]

Lol! I am teaching a calc2 class in the fall. We will see!! : )

[@JorgetePanete]

blackpenredpen the next year i will see integrals on our tests and i need a bless from Pitagoras to pass these exams

[@Metalhammer1993]

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)

[@hendrickdias8366]

Watching this made me very relaxed. You sir deserve a subscribed!

[@Far_Gem]

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).

[@MC-mx1mt]

Florian Bender if you mean 1/tan x, he would write it as cot x

[@monzur1947]

Florian Bender tan^-1 is a standard notation for arctan

[@GrouchierThanThou]

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.

[@angelmendez-rivera351]

Monzur Rahman We're aware, but it's a terrible notational standard.

[@GrouchierThanThou]

It's not even a "standard" notation really. Just a commonly used one (and not even as commonly used as the arctan notation).

[@davidgillies620]

It's also expressible in terms of "the" hypergeometric function 2F1.

[@clementtan5004]

here i am watching 30 minutes of math porn and not knowing wtf is going on

[@kibayuuto6634]

Clement Tan isn’t that the case when watching regular porn too?

[@suyunbek1399]

Жиза

[@aryanjha1302]

Easiest integral on KZbin 31 min yt video - "no i don't think i will"

[@vj_henke]

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...

[@blackpenredpen]

oscarjd74 OH WOW!!!! YOU ACTUALLY DID IT!

[@vj_henke]

NEAT, instant screen saver ^^

[@caducoelho2221]

How cool iz dat?! HAHAHAH

[@omarifady]

I have curiosity of what you typed on the keyboard at 27:21 ...

[@AviNash6817]

Good. this is a high level problem in my sheet .

[@NPC-rq9tp]

At 18:00 , why not just use trig substitution. I tried it using 3^(1/2)/2 tan theta. It worked out alright.

[@benjaminbrady2385]

I was kind of hoping to see you dancing in celebration chanting "One Take!" at the end

[@xamzx9281]

who knows maybe its 10th take

[@blackpenredpen]

Notice that I didn't edit this video!

[@markzero8291]

MrQuantum "one take" and "1st take" are different.

[@blackpenredpen]

Mark Zero true. One take for this tho. I think it was my 3rd take bc my intros weren't good enough.

[@xamzx9281]

Mark Zero k

[@jihanhamdan5465]

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

[@sunglow9835]

[@blackpenredpen]

: )))))

[@AdamMajewski-h1h]

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

[@sirfactor]

You earned yourself a new sub Thanks sir .

[@blackpenredpen]

Glad to hear! Thank you!

[@meraldlag4336]

I'm so happy to see a "BIG INTEGRALS NO SHORTCUTS" playlist, will watch them all

[@ulilulable]

This is so great! I wish I still had the "mathematical muscle" to do this myself. Maybe one of these days...

[@thelegendofme7520]

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

[@ulilulable]

@@thelegendofme7520 Aaany day now... ;)

[@mrstutoring]

very very good video and very satisfying to have a clean solution in the end.

[@ronnykazadi559]

Can you do integral of (xtanxdx) ?

[@Fematika]

I did the differentiation! Find it here! kzbin.info/www/bejne/l6SUp55mftp0fqM

[@JBaker452]

This does turn out to be a monster of an integral "doesn't it" :-)

[@blackpenredpen]

Yea, isn't it?!

[@JBaker452]

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).

[@Harothir]

This is just what I needed for my summer.

[@Zhardamon]

pure beauty...!

[@blackpenredpen]

Thank you!!!!

[@klokcotizao224433199]

And my professor is expecting me to solve that in only 10 minutes.

[@citizenalex1934]

"When in doubt, use horseshoe mathematics."

[@craftbuzzwonky4752]

You too saw the video!! I see, you are a man of "NO U" culture here!

[@vynderma]

Glad you enjoyed yourself with that. This is a great review of a lot of techniques.

[@Y650-Y]

It is too long, not too hard. Anyway I doubt anyone can do it in any exam without making a small mistakes ☺

[@blackpenredpen]

it's like running a marathon (well, almost...)

[@DougCube]

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.

[@LightYagami-on5lm]

When title says easy and time says half an hour 🤔🤔🤔🤔🤔🤔🤔😱

[@arnavanand8037]

Cringe

[@NWSCS]

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.

[@LucBogaert]

Running a marathon is easy compared to solving this integral 🤓

[@KrisEditz29]

Most elegant integral I've seen so far,thank you sir

[@ktuluflux]

Nothing even remotely elegant about this. This is the equivalent of opening a safe by dropping an anvil on it.

[@roceb5009]

Okay so you've done sqrt tan x, you've done cube root tan x, can you generalize? Nth root tan x ?

[@Abc-tx4zr]

Lost at 20:30 would be glad to here an explanation of that bit, the splitting the fractions and the arc tan part

[@Bludgeoned2DEATH2]

Dr. BalckPenRedPen, please check the answer by differentiation :)

[@prollysine]

Gratulálok, ez a szinte végig helyettesítéses módszer bemutatta, hol kezdődik az elméleti matematika !

[@TheKillerSock]

yea dog imma need you to check by differentiation

[@enriquevanegas5741]

Jesus Christ! With this you have won a new follower! Subscribed!

[@thewatcherinthecloud]

...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

[@xunxekri]

13:12 Feels like quark electric charges.

[@Centauri902]

now the nth root

[@nope110]

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.

[@maalikserebryakov]

dude same 😭 To think i can finally keep up with some videos on this channel

[@capjus]

You can't impress a girl with it

[@bmc868]

sure it's too smart for them....

[@timstorer978]

I disagree. Perhaps your trying to impress the wrong girls.

[@azmath2059]

No you can't impress the girls quite true, but you can impress them with the money you make if you're working in this particular field.

[@General12th]

If you can't impress your girl with this stuff... *you're not talking to the right girls*

[@bmc868]

Almost all girls like money, luxury cars and spending time shopping. They want M-O-N-E-Y......and their math skills is +....+money+money+money......

[@speedrunschroedingerscat96]

"Easiest Integral on KZbin" - *31 minutes long*

[@robertherbert8419]

....It wasn't that bad....it didn't have limits to substitute (hehe ! )

[@BRUBRUETNONO]

Tumbs up for this ace demonstration !!!

[@davidkippy101]

Please check the answer by differentiation

[@laxminarayanbhandari855]

My friend had given me this integral as a challenge and I had solved it. I would say this a great algebraic integration problem.

[@jasoncetron233]

Small technical error: In the final function, you used the constant C to represent the constant of integration. BUT, you already used C to represent a different constant in your partial fraction decomposition. So, the final solution should use a different constant, like C-sub-1 or K.

[@TheTimeDilater]

seems like a deep breathtaking moment when you got the answer to be this long...