Sorry for the reupload. I actually had a mistake on the integral of 1/(x^2-a^2) in the previous video. I will make up to you guys by checking my answer by differentiation! That video will be done soon!

@zackthotho7459

6 жыл бұрын

Np. it is fine

@lxathu

6 жыл бұрын

This one's more worth the storage area of google than a dozen 9 percent of the content of YT.

@KeyMan137

6 жыл бұрын

blackpenredpen Thanks

@AhnafAbdullah

6 жыл бұрын

Did you have to redo the video? or just edit the definition that showed on screen

@subinmdr

6 жыл бұрын

How do we factorize (u^4 + 1) ? plz

Imagine doing all this and then forgetting the +c

@actually_2337

5 жыл бұрын

Hamish Blair lol

@shayanmoosavi9139

5 жыл бұрын

You would get zero marks in exam😂😂😂😂

@yrcmurthy8323

5 жыл бұрын

Edit the video

@triviumfanmexico

5 жыл бұрын

It actually happened to me in an exam :(

@user-ft6sw7wy5k

4 жыл бұрын

According to the question. For example, if the question is to find an original function for the next function, the answer without the constant is correct. But if the question is assigned to all the original functions. It must make +C

you know something is hard when blackpenredpen uses five colours

@seshnarayan7972

3 жыл бұрын

I only noticed 4 colours

@spooky2526

2 жыл бұрын

@@seshnarayan7972 blue black red green purple!

@createyourownfuture5410

2 жыл бұрын

@@spooky2526 Where is purple?

@esajpsasipes2822

11 ай бұрын

@@createyourownfuture5410 he mentioned he uses purple for the second part of observe section (10:39) although it doesn't look that diffirent from blue

I think my professor summed up integration in a nice way. He said differentiation is all about technique. You see a scenario and have a set of rules you then follow. Integration is a form of art. It's much more intricate and delicate.

@blackpenredpen

6 жыл бұрын

TheDaltonGillespie I totally agree!!! Has he done this integral with u guys?

@agmt233

5 жыл бұрын

My Further Maths teacher calls integration Black Magic. Two kinds of people I guess

@marcushendriksen8415

5 жыл бұрын

Integration is definitely more difficult, and thus satisfying

@tommyron1792

4 жыл бұрын

I agree. It takes a clever mind to do differentiation problems with ease. But it takes a creative mind to do integration problems.

@shoobadoo123

4 жыл бұрын

AGMT further math? Are you international baccalaureate?

"1+1 is 2, right?" Calculus, everybody.

@blackpenredpen

6 жыл бұрын

Yes. But in Z2, 1+1=0.

@blazep5881

6 жыл бұрын

well 1+1 is 1 if you account for boolean algebra :)

@ganaraminukshuk0

6 жыл бұрын

You know you're high on math(s) when you forget what 1+1 is.

@aronquemarr7434

6 жыл бұрын

1+1=10 in binary

@guisilva9815

6 жыл бұрын

1+1=3

"Welcome to the Salty Spitoon, how tough are ya?" "How tough am I? I just integrated a trig function!" "Yeah, so?" "integral(sqrt(tan x))dx" "Uh, right this way..."

@megauser8512

4 жыл бұрын

Lol

@TheLucidDreamer12

3 жыл бұрын

"I just integrated 1/(x²+tan(x)) dx"

@jakepfeiffer6577

2 жыл бұрын

∫√t̅a̅n̅x̅ dx

@createyourownfuture5410

2 жыл бұрын

@@jakepfeiffer6577 WHERE. DID. YOU. GET. THAT. INTEGRAL. SIGN. FROM?

@jakepfeiffer6577

2 жыл бұрын

@@createyourownfuture5410 on Mac it’s just option + b Idk about windows but you can copy/paste it

Mr Math Man. Math Me a Man. Make him integrate the square root of tan.

@kastormorgan1536

5 жыл бұрын

stealing my joke, still love you though babe

Me when finding this channel: "wtf is going on?!" Me rewatching 1 year later after having seen every bprp video: "alright easy didn't even need the DI setup"

this is pure game. There are very very few maths teachers at level of you. Thank you.

Integration is just the easiest thing ever... I can integrate √tanx + e^x² in seconds: Set up the integral: ∫ √(tan(x)) + e^x² dt And then just use the "inverse" power rule: (t)√(tan(x))+ (t)e^x² And we're done... I didn't say that I'ld do it with respect to x...

@restitutororbis964

5 жыл бұрын

Viktor Sundström LOL, or you could use horseshoe integration, Indeed one of the most powerful mathematical tools out there.

@neilshah754

5 жыл бұрын

Still forgot the +c dude 😂

@cesarturanzasfarill2976

5 жыл бұрын

Sólo los pendejos dicen que está facil resolver un problema.

@gbugis6706

5 жыл бұрын

@@neilshah754 *depression intensifies*

@pablovirus

3 жыл бұрын

@@cesarturanzasfarill2976 calmate viejo, era un chiste. No lo entendiste??

Wow..you demonstrated this before 100k subs more than 3 years ago, today you have 687k. Wishing you for the next 313k

@blackpenredpen

3 жыл бұрын

Thanks

@parkeryoung2471

2 жыл бұрын

@@blackpenredpen but what is special about the number 313 ?

@prakharjain21

2 жыл бұрын

@@parkeryoung2471 it's prime

@joshuacobblah364

Жыл бұрын

@@parkeryoung2471 to make it a million subscribers

Evil integral to place on an exam ... :/

@rawn4203

4 жыл бұрын

Would have to be like the only question or 1 of 2 questions.

@kishorekumarsathishkumar1562

4 жыл бұрын

I'm from India, and i am practising for this exam, I can assure you, there are more brutal questions.

@rahimeozsoy4244

3 жыл бұрын

@@kishorekumarsathishkumar1562 yeah for Indiana this is ez or normal

@sgsnake2x

3 жыл бұрын

@@kishorekumarsathishkumar1562 im guessing they give you the space because with an A4 blank paper this would quickly turn messy for me as my handwriting is pretty big

@Wu-Li

3 жыл бұрын

This one is an easy problem

I'm preparing a transfer exam for Korean universities and there was this question on my preparation problem set. Your solution was so helpful brother, thanks a lot!

@zohaibgulawan9070

Жыл бұрын

And what you doing? In uni?

@user-gr5lr7sm9e

Жыл бұрын

@@zohaibgulawan9070 Electrical and Computer Engineering, but I think I failed the exam. I might just quit and make indie game

@alfredomulleretxeberria4239

9 ай бұрын

@@user-gr5lr7sm9e So...how did it go for you?

I passed my semester of Calc 1! I did not fully understand everything but I believe I got a strong majority of it & I will be working on some of my weaknesses during break to prepare for Calc 2. For some reason when we got to U-substitution, everyone was confused but it seemed to make sense to me just based off the few example she gave in class and somehow that was all I needed. Anyway, so during last class before our final, she did a little review & answered questions. Someone asked about this sqrt(tanx) and she was like "oh you cant solve this with your current toolset, wait till next semester" and I was like "I can solve it" because I had all this false confidence from having understood the whole U-sub stuff. Well turns out this is a VERY difficult one lol. I guess if you are a student and want to be prepared for integrals, just spend a week studying just THIS one integral and you should be good to go lol.

15:02 WIZARD! Where can I buy this magic blackpen??

@agfd5659

6 жыл бұрын

WTF! I didn't even notice it before!! How did he do that?!

@morganmitchell4017

6 жыл бұрын

He edited it because he forgot the (-) sign

@weerman44

6 жыл бұрын

Haha I understand that Just a silly comment ;)

@vvsutar6179

6 жыл бұрын

Morgan Mitchell Johnson

@habiburrehman7108

6 жыл бұрын

hahaha i also noticed after you comment.....

14:41 "This is the two, so what should I do?" What a rhyme :O

god damn I love math

@blackpenredpen

6 жыл бұрын

Me too!!!

I just saw you instagram reel, where you give credit question and searched for this integral on KZread. And I am so glad I found your video 😀!

@blackpenredpen

2 жыл бұрын

😆

Ok, so in the beginning we have a pretty simple mathematical expression while the result in the end is something horrible. Things tend to evolve naturally from more complex to simpler. I therefore consider it normal to leave this formula unintegrated. Thank you for all your thumbs up ! :D

When you do all of this in the exam and you realize the integral was sqrt(tanx + 1)

Very well work dude, the resolution was easier than I thought, I had problems when using trinomio. You got a like and a new subscriber!

You, my good sir, are turning calculus into art! Awesome video!

Oh man, I love this video. I watched the entire thing and enjoyed every second of it! Keep up the good work on your channel. :)

@Someone-cr8cj

6 жыл бұрын

DGCubes what are YOU doing here??¿

@DGCubes

6 жыл бұрын

What can I say, I like calculus. :P

@blackpenredpen

6 жыл бұрын

thank you DGCubes!!

@metalmathprofessor1467

6 жыл бұрын

Me too, I watched the whole thing wondering what was going to happen next! Really a great calculus problem. I'm going to show it to all of my students!

@thephysicistcuber175

6 жыл бұрын

BOI!

Simple presentation of the difficult integral in a nice manner . Thanks .

I like this integral and its anti-derivative. I am impressed with your technique of using trigonometric and u-substitution along with algebraic manipulation to arrive at the answer.

The most difficulty integral that i ever seen in my entire life! But it was really good xD

@blackpenredpen

6 жыл бұрын

Thank you : ) And..... there's the integral of cbrt(tan(x))

@raytheboss4650

Жыл бұрын

@@blackpenredpen imo that’s easier than this

You made me fall in love with mathematics!❤️ Thank you!

I LITERALLY LOVE YOU SO MUCH YOU DESERVE THE WHOLE WORLD

I majored in physics in school and always preferred the more pure abstract mathematical parts of it. Watching this video is like taking a mental vacation back into the past. I'm happy that I was able to follow it through to the end on my first viewing :)

Gratulálok, lenyűgöző végig az átváltások sorozata ! Főleg az (U^2 + (1/U)^2) átalakítása !

10:54 defenitely most important part

I always understand your calculus explanations. Thank you.

10/10 what a trip

@blackpenredpen

6 жыл бұрын

Yea, I know!

Your reslly entertaining and it is very interesting. I love your out of the box thinking to manipulate things to make them work!,,

Ez annyira bonyolult, hogy nincs értelme ennyit számolni, de blackpenredpen igazi zseni !

I just differentiated this myself, it starts out ridiculously complex, but it slowly starts to fit in with everything, good luck on doing it! You might need 6 boards to do it, I managed to do it in 1 board, but I had to rub out a lot of the work out I did

@ramking7869

Жыл бұрын

Differentiating sqrt(tanx) is not hard at all😂

@RayTracingX

Жыл бұрын

​@@ramking7869 he is about differentiating the antiderivative of sqrt(tanx)

Got to see this question first time on my exam today... carrying weightage of 6marks.. was totally fucked up😑

@kava2214

3 жыл бұрын

Which exam?

Riveting :) Bravo on a brilliant solution! I think I could explain that to someone else now!

Slightly more straightforward (although much longer/messier): Factor x^4 + 1 = (x^2 + sqrt(2)x+1)(x^2 - sqrt(2)x+1), then do partial fractions, complete the square in the denominators and solve. This saves you from having to figure out the trick where you add 1/u^2 and subtract 1/u^2.

Fantastic video. Very well explained. Thx

And pretend nothing happened!?? Great lines

Man, so many colors! If you wrote blackpenredpen in that empty space and taken a picture, you'd have a pretty good channel banner.

Love your videos sir, you make very complex calculus part easy 😃😃

This is lengthy problem. Very few can solve in first time. We can only solve this problem if we practice at home. Your explanation is very nice

Simply great! Thank you a lot master.

you are seriously the best maths teacher!

So... u substitute for fx Square u = sqrt (tanx). See that 2udu = sec^2(x) dx. Squaring u^2 = tanx gives us tan^2(x) = sec^2 (x) -1 = u^4 which we can see as sec^2 (x) = u^4 + 1. Thus dx = 2udu/u^4 + 1. Plug in original equation to have integral of u*(2u/u^4 + 1)du. Multiply top and bottom by 1/u^2 to get complex fraction with sum of squares in denominator. Complete the square to get (u + 1/u)^2 - 2, which has derivative of inside equal to 1 - 1/(u)^2. Now we want two integrals (why? it is not clear unless you see the tanh^-1 and tan^-1 option coming up), one with 1 - 1/(u)^2 in numerator and other with 1 + 1/(u)^2 in our numerator. Because the completed square can have two forms we can have the appropriate denominators to do two more substitutions, this time with t and say w. If we do the substitutions correctly we have two integrals, one being of 1/(t^2 - 2), and our other being of 1/(w^2 + 2), both in their respective worlds. A formula exists for these forms to be integrated neatly into tanh^-1 and tan^-1 forms. Substitute u back in for t, w, and sqrt (tanx) for u. Do this correctly, and then Add c and we're done. I did this mostly for my own understanding, but I'm fairly sure I didn't skip too much for it to act as a quick summary.

@blackpenredpen

6 жыл бұрын

This is great! It's good to work out the problem on your own or along the way.

10:43 that's your brilliancy sir

Very good explanation... Thanks a lot

Did you guys notice how he changed his pens at 05:23? That was awesome!

Great explanation! Thank you! :)

you are awesome men this appear on my exam from yesterday i lov u 💕

Finally! Loved the video! Is great! You could do the same integral for the 100k, but now you do it the hard way! (The one with partial fractions and with the factorization of u^4+1, and with the natural log result!)

@blackpenredpen

6 жыл бұрын

Mauro Castañeda I would need another white board for that tho. Lol

Watching for second time, now i got it! :D Great video!!!

@blackpenredpen

6 жыл бұрын

yay!! Thank you!

Yesss! This question was on my calc 2 final exam and I got it right!

You are a brilliant Math teacher

Gracias, haces todo fácil Saludos desde Colombia

sqrt(-1) just love your videos! I'm gonna start studying math very soon and your videos really hype me for it^^

@blackpenredpen

6 жыл бұрын

THANK YOU!!! I AM VERY HAPPY TO HEAR THIS!!

@agfd5659

6 жыл бұрын

Your pun made me cringe. I hope you're happy! :D

@mrocto329

2 жыл бұрын

@@blackpenredpen 4 years later you are still inspiring people. I'm only 14 right now, so don't have any solid plans for uni etc. (other than studying CS as I like programming) but now you got me interested in maths! I've been spending my afternoons just trying to learn maths for the past few weeks, and it's been really fun so far!

It feels nice that i solved it all by myself for the most part. I have some amazing teachers. Can't thank them enough

Thank youuuu❤, greetings from 🇨🇴

@blackpenredpen

4 жыл бұрын

Thank you! :)

Absolutely amazing 👍🏻

6:13 out of context is beautiful

I couldn't do this integral without u

We can solve this in some other way too. We can write sqrt tanx as 1/2 {(sqrt tanx + sqrt cotx)+ (sqrt tanx - sqrt cotx)},and then break these into sin cos expressions,and then subtitute sinx + cosx = t and sinx-cosx = k in the first and second integrals respectively,and then apply the standard integral formula. Anyways love you process too!

Nagyon jó és elegáns megoldás !

19:05 Try to differentiate THIS to give sqrt(tan x)

YAY I got this correct Imma do a video on how I did it and then compare it with your method. This took me ages btw

Great explanation!👍

prolly the first time im actually being happy after a maths answer

What a gorgeous way to find the answer! Who is the genius first to find u+1/u and u-1/u pairs for this question? It really drive me crazy for such an integral. Thank BPRP.

Cool to see it done in an alternative way :)

Best birthday gift , thanks

Couldn’t you have just done tan inverse with the original u equation after you divided the fraction by u^2, or does it only work with one variable and one constant term?

@alejrandom6592

Жыл бұрын

No, cuz then you wouldn't have the +1/u² term nor the -1/u². You need them both to cancel each other.

You could've used another formula/ method for integrating the 1/(t^2 -2) just by using : integral of 1/(x^2-a^2) = (1/2a)*( ln( |x-a|/|x+a| ) . that would have been more easy and intiutive than hyperbolic inverse function.(just saying) BTW very nice explanation sir. Great content. You're a wonderful teacher. PEACE

Amazing explained

The math is a universal languaje. I'm peruvian, but i can see this video and understand it!🤩

I've seen this video a couple years ago but decided to comment only now. First of all, very nicely done! Without trying to diminish guy's effort and all the excitement of the viewing public, I just wanted to remark that from the view of pure math this is absolutely worthless. Here's why: In all applications (including physics, engineering, and even math) ALL integrals are definite. This integral would be of interest only if integral is from, say 0 to pi/2. In couple of steps now I'll solve the problem of integrability and the value of that integral. First, simple substitution v=pi/2-x translates this integral to the integral of \sqroot(cotv) over the same integral. Second, cotv is approximately inverse the of v for small v, so the question is \sqroot(1/v) integrable, and the answer is, of course yes. And we are done! If somebody needs the numeric value, just take integral of \sqroot(1/v) from 0 to 0.01 and then calculate the remaining part from 0.01 to pi/2 by whatever method of approximate integration.

I think that the most monumental achievement humanity could ever hope to accomplish would be to find an intuitive understanding of how to go directly from the integrand to the antiderivative in one step...

This turned to blackpenredpengreenpenbluepenpurplepen real quick

@blackpenredpen

Жыл бұрын

😆

Just... beautiful !

@blackpenredpen

6 жыл бұрын

Thank you!!!!

The hardest part about this is to remember the +C.

I guess I'll comment again. You could do e^(-x^2) for a 100k!

@blackpenredpen

6 жыл бұрын

Lol! I hope that's from -inf to inf

@SoumilSahu

6 жыл бұрын

Wow! I can't believe you remember!

@blackpenredpen

6 жыл бұрын

Soumil Sahu I do!! Lol

@blackpenredpen

6 жыл бұрын

Soumil Sahu I am kinda sad since i lost all the wonderful comments from the previous video. But I really wanted this to be fixed

@holyshit922

6 жыл бұрын

Error function, Gamma function , series expansion, nothing else can be done

I solved this problem long back. ie in 1982 when I was in class 12th.

If you plug in 0.5 for the integral function, then (sqrt(tan 0.5)+sqrt(cot 0.5))/sqrt(2)= 1.4793, but arctanh(1.4793) is undefined. The range of (sqrt(tan x)+sqrt(cot x)/sqrt(2)is alway greater than 1, which makes arctanh(sqrt(tan x)+sqrt(cot x)/sqrt(2)) always undefined for this integral function.

@DougCube

6 жыл бұрын

undefined or complex, but either way I think you are right -- see my comments

@williamwen7190

6 жыл бұрын

DougCube But integral from 0.2 to 0.5 of sqrt(tanx) is defined and real from the graph of sqrt(tanx). And all the result of this integral function is complex or undefined.

As my Cal 3 Professor used to say " What could be simpler ."

Very symmetrical so much so with all those arctans and tan x's you could combine them except for that little h in the inverse hyperbolic tangent it stands for Hell in this integral bc you can't take tanh^-1(n) ||n|| > 1, ~|tan x| + |cot x| w/o screwing up the rest of it unless it's some kind of cubic solution with complex b coefficient ~ 1/3 ln (i) Point it seems so close to y = f(x); f^-1(y) = x, x could be 0-2π with no trouble you would run into complex numbers but they cancel themselves out ish

One rather interesting thing about that expression is that (if we take C = 0) we'll generally get a complex number out of it. The imaginary part is always the same though. I think you get a nice expression if you introduce complex numbers (starting by factoring (u^4 + 1) into (u^2 +- i). But it seems non-obvious that 1) the manipulations I did after that are totally ok, since I kind of ignore principle value etc... and 2) how to rigouously show this is actually equal to the answer found with real math. Writing R = (1 + i) / sqrt(2) and u = sqrt(tan(x)) (and Re == real part) then I eventually got that Re[2R*arctan(Ru)] is an antiderivative of sqrt(tan(x)) w.r.t. x Writing out the def. of (complex) arctan => Re[(1/R) Log [(R - u) / (R + u)]] is an antiderivative. Numerically checking for some random values I find this does give the same values (up to a constant) as your solution. Though I did assume that tan(x) is non-negative here (that is how I ended up with "real part" in there).

was trying to do on my own but was stuck after first substitution this video helped.

He didn't reupload this video, we all just have a déjà-vu at the exact same moment.

@blackpenredpen

6 жыл бұрын

Sorry for the reupload. I actually had a mistake on the integral of 1/(x^2-a^2) in the previous video. I will make up to you guys by checking my answer by differentiation! That video will be done soon!

I liked it very much!

The trick is to complete the square. Got it.👏👏👏👏

Hey I Am JEE aspirant Thanks bro 😁

Every integral tackled for the first time is a journey into the unknown...

Hey bprd I have suggestion in 3:31 , we can take u^2 as t and then do partial fraction , it will be easy , after doing partial fraction substitute t as u^2 .

There is simple method (I Feel) put x=(pi/4-t) dx=-dt and integral become sqrt(1-tan(t))/1+tan(t)) after rationalising numerator we get (1-tan(t)/sqrt(1-tan(sq)t) which is (cos(t)-sin(t))/sqrt(cos(sq)t-sin(sq(t)) now split the integrals. Cos(t)/sqrt(1-2sin(sq)t) other integral is sin(t)/sqrt(2cos(sq)t-1) by sqrt(2) manipulation we get u/sqrt(1-u(sq)) other u/sqrt(u(sq)-1) both forms are familiar having formulae

Love this channel

@blackpenredpen

6 жыл бұрын

Travis Hayes thank you!

3:50 take t = u^2

I did the integral in a slightly different way. I ended up with (sqrt(2)/4) * log(abs((1/2 + (sqrt(tan(x)) - sqrt(2)/2).^2) / (1/2 + (sqrt(tan(x)) + sqrt(2)/2).^2))) + (sqrt(2)/2) * atan(sqrt(2*tan(x)) + 1) + (sqrt(2)/2) * atan(sqrt(2*tan(x)) - 1). I also let u = sqrt(tan(x)) but in of multiplying through by 1/u^2 I added and subtracted 2u^2 from the denominator and then factoring the result using difference of squares. Once I had it in a factored form, I used partial fractions.

You're awesome man thank u

the sheer joy with which my man just said: "I also have a purple pen :)"