checking the answer of integral of sqrt(tan(x)) by differentiation

[@tylerqueen8920]

It's just so beautiful...*tear falls slowly from eye*

[@blackpenredpen]

Tyler Queen you won the comment of this video!! I just pinned it!! What a wonderful comment!!

[@pongwang9893]

@@blackpenredpen yeah , so brilliant . great

[@AP12uZvfvag]

Last time someone used chain rule so many times, slavery was still legal.

[@afsarshaikh9654]

AP12uZvfvag You made me laugh. thank you.

[@megauser8512]

lol

[@BizVlogs]

ummmm… THE FUCK?!

[@donkeybros8734]

I’m dead 😆 😂

[@erikmensinga]

No its the chen lu

[@stuartyellow1679]

Why you dont call the video "differentiat (-1/sqrt(2)*tanh^-1((sqrt(tan(x))+sqrt(cot(x))/sqr OK i got it

[@blackpenredpen]

exactly

[@diegobatista2239]

latex gang

[@bastianherrera2171]

I've watched most of your videos and they are amazing, mainly because of the clear way you solve the problems. I've learnt so much about calculus with you, thank you very much! Keep making these amazing videos!

[@blackpenredpen]

Thank you for the supporting message! Will do!

[@morganthem]

blackpenredpen I second this, they are amazing and clear

[@AhnafAbdullah]

I did this derivative after he put out the sqrt(tan x) but before this video, it was so beautiful :-:

[@JeronimusJack]

oh yea that's excatly how i feel, too. my old math tutors always wanted me to break more about it instead of telling me how it's done, but sometimes i was just stuck, here i get answers. i did the derivation afterwards myself :D and it worked. it's better to learn this way since i can explore and my stress level stays down hehe. it's not torture but tutoring here. ty

[@joelottosson1297]

You like when they cancel out don't you

[@blackpenredpen]

sure i do!

[@ZoneEEEEEEEEEEEE]

It's just so wonderful, it's almost like magic :')

[@whiz8569]

"Don't be intimidated cause I did 1+1=2 again." I see what you did there.

[@blackpenredpen]

whiz 85 hahahahahahaha!!!!!

[@throxs1535]

I feel like you're really getting the hang of this now :) going slow where need be... Going faster where allowed... This all comes together in a very fluent yet easily understandable explanation :) keep it up!

[@blackpenredpen]

Throxs MC thanks!!!! I think it's bc I have been just doing higher level math lately so it was easier to get the pace right.

[@DancingRain]

That was epic! The most amazing mathematical simplification I've ever seen.

[@turkishwagnerian]

"I chose mathematics to honor the human mind" Carl Gustav Jacob JACOBİ

[@morganthem]

Wow

[@mrbenwong86]

After 15 minutes, I forgot I am doing differenitation, I almost want to put +c at the end.

[@Magic73805]

Really, You're great sir. You are the master of "Calculus".

[@furo822]

I think this is better. For the 1/(tanx + cotx) Multiply both sides with tanx ->tanx/(tan^2(x) +1) and the bottom part becomes sec^2(x) Finally everything just cancelled out and sqrt(tanx) remains That was funny btw

[@blackpenredpen]

YungHong Kung ahhhhhhh that's better!!!

[@agabe_8989]

That beast was too long that he had to type "integral of sqrt(tan(x))".

[@איתןגרינזייד]

I really love how it all just collapses down to sqrt tan x

[@kornelviktor6985]

Its so nice to see that he gets very excited when he sees that the answer will be sqrt(tanx).

[@tcocaine]

I love how in the thumbnail you just wrote craziness instead of the whole shablang. I absolutely adore it.

[@blackpenredpen]

HAHAHA, thanks!!!!

[@soyoltoi]

tanx for the video

[@dizilloHT]

wow finished my calculus class this year, and i dont believe i will see it again, but wow im so amazed by it. Keep up with the videos you help a lot of people with them!

[@flowingafterglow629]

So I watched the integral of root(tan x) the other day, and therefore had to watch this. I like the way you were smiling through the whole thing. I was just laughing the whole time, it is so beautiful to see. Every time something would fall out, I would just laugh some more. Amazing.

[@shiina_mahiru_9067]

at 10:24, instead of changing tan x in terms of sin x and cos x, I would just multiply sqrt (tan x) both upstair and downstair to the second fraction. That should be the first thought because you want to end up with sqrt (tan x) after all the simplification, and to make it happen, sqrt (tan x) must be in the numerator

[@Sid-ix5qr]

d/dx(craziness) Let's do L-sub. l=cr d/dx(laziness).

[@ttg001]

I'm holding back tears....... This video is so powerful

[@MRBYRNO2505]

holy mother of god, if i saw this a few years ago i would not have chosen to study maths at Alevel. seeing it now just brings a tear to my eye its so beautiful

[@snejpu2508]

0:05 "We'll be differentiating this chinese expression". : )

[@buckeye49]

He said giant

[@BlokenArrow]

That’s wasist

[@drewski91]

i started crying at the sheer beauty of this math

[@weerman44]

Yet again, an amazing video with such a clear explanation

[@Xeotroid]

1a) Integrate sqrt(tan x). 1b) Check the answer of 1a by differentiation. 1c) Prove 1b.

[@obinnanwakwue5735]

Let epsilon > 0...

[@MurakDurak]

integral of sprt(tanx)dx has to be my favourite integral, but differentiating it is even more crazy!

[@dakotaroberson9921]

Wow great proof!! it was so great to see it all come together at the end lol

[@blackpenredpen]

I know! It was so satisfying!!!

[@tomatrix7525]

That is a beautiful piece of art my friend

[@AlmightyMatthew]

just perfect. satisfaction at the end was through the roof

[@Galileo2pi]

Great, keep going please

[@bsul03420]

Excellent, but now I have to differentiate my inverse coth expression to show it's the same!

[@gvchas]

Everything is possible. Just believe in your math!

[@MarkMcDaniel]

Brutal, but satisfying as hell.

[@helved807]

this is the most satisfying differentiation

[@Lily-zd6dx]

Mindblow

[@TheYoshi463]

Yay, blackpenbluepenredpen made a new video!

[@blackpenredpen]

I did yesterday too

[@bartj777]

Bravo, sir. Very impressive.

[@azmath2059]

That's a great deal of work! Thanks for posting. Brilliant

[@fakeplaystore7991]

This is evil sorcery, pure and simple.

[@axel15473]

He is so on fire XD

[@elementalsheep2672]

from about 12:00 onwards you could hear him getting more and more excited

[@cheeragsridhar1696]

It’s so awesome The rate of terms falling from my eye is differential

[@dyer308]

Please show us Feynmans trick on integral solution of sinx/x without a power series solution!!!!!!

[@calcul8er205]

raees khan Another cool way that uses the idea of multiplying the integrand by e^(-st) is by double integral and Laplace transform

[@Justiin_rm]

Woaaaa

[@berenjervin]

Great video. Makes me want to go back to uni for a math degree after I retire!

[@MrPhilipp456]

This is insanity! I love it

[@BRUBRUETNONO]

Ace derivative master ;)

[@wydadiyoun]

How about starting from square tangent and find that expression by integrating

[@eetuhaavisto524]

Hey blackpenredpen can you solve this derivative next time with the definition of derivative? That ought to be an interesting limit(needless to say that very fun as well)😂

[@blackpenredpen]

I prob. will have to use 4 classrooms for that...

[@samirgeiger1042]

This is just so amazing. Great work! Such beutiful derivatives

[@johnsalkeld1088]

At the stage of getuve rid of tan plus cot recall that this is cot * [1+tan^2] so cot * sec^2 easier I think due to direct cancelation and resolving the full Eqn to 1/(cot * tan^1/2 ) or tan ^1/2. but from a pedagogical standpoint I see that using a standard method is good

[@bhavydugar6665]

This is just so coooool I took its SS and made it my wallpaper

[@alomirk2812]

Just wanted to point, this is the solution of the integral but it is undefined in the reals, if you will extend it to the complex numbers it will be defined and give you a real number

[@JulianOnions]

This is amazing!

[@IISH4RKZzII]

This is SOOO DOPE !!!

[@PackSciences]

Valid forall x in [0;pi/2[ mod 2pi. You could have made it more impressive using the formula arctan(x) + arctan(1/x) = sign(x) * pi/2 in order to get an uglier expression.

[@blackpenredpen]

LOL!!! maybe next time

[@id120266737]

You're a fucking genius.

[@TheScienceGuy10]

Could you please do a video of integration of sqrt tanx? The other way round. I've been trying it out but getting stuck. Would really appreciate it.

[@jmannthemann4793]

Siddhartha Pundit He already has, just look for it

[@osvaldomena1172]

Amazing man

[@blackpenredpen]

Thanks!!!!

[@ck7671]

8:19 simplify the 2/sqrt(2)^2 !!

[@XXgamemaster]

Beautiful.

[@July__-eu5pd]

Wow you actually differentiated it :p

[@garomo6384]

amazing !!!!

[@ExiDJuicy]

Missing "+C" at the beginning 😎

[@OonHan]

DIFFERENTIATION DOESN"T NEED A +C

[@unknownanonymous5914]

+Oon Han thejoke.png

[@arekkrolak6320]

If you have to remember derivative of arc tanh then you can also remember what id the final result :) be a real man and use definition of derivative! ;)

[@vibhupandya6103]

Hey in sqrt cotxtanx won't it be = sqrt1=±1???

[@Alisssap]

Waw

[@АндрейПетров-д2э]

Miraculo!

[@wellbangok8959]

I'm a simple guy. blackpenredpen uploads a video, I watch.

[@blackpenredpen]

Sheev Palpatine lolll thank you!!

[@imcalculus]

Do this with definition of a derivative

[@upanshulakhani6221]

Now check the answer by integrating

[@golocok]

Un final de cálculo que te haga resolver la integral de la raíz cuadrada de la tangente y luego derivar para demostrar que el resultado estaba bien. No aprueba nadie. :D

[@colleen9493]

Now to the integral of the integral of sqrt(tan x)

[@ReubenMason99]

Could you possibly do some more detailed explained integration videos for people who don't know much calculus

[@francoisgatine4106]

There is a youtube channel called Professor Leonard (hope I wrote it correctly) who records himself teaching several maths classes. He's a really good teacher in my opinion, it can help you with understanding calculus. No, it isn't boring.

[@blackpenredpen]

Reuben Mason hi. U can also check out my playlists. www.blackpenredpen.com/math/Calculus.html

[@kindlin]

That's a detailed list. How did you have the time to put all that together with all your regular classes/study etc.?

[@majkgmajkg2613]

I'm sorry early too... In my country is 0:14 am. On 7 I have job... Shit...

[@AdrianMartinez-lt4dz]

Man! You're a genius 👌 Now do it in chinese xD

[@barbarodave]

Why?

[@afsarshaikh9654]

Mind Fucking Blown. Just see how beautiful that is!

[@Kasual_1]

Integrate your answer again to double check

[@ToeNailMuncher111]

How does he do the chain rule so fast

[@chessandmathguy]

beautiful. just beautiful.

[@axel15473]

Oooohhhhh yessss. I wait for that. The other way

[@stolenmonkey7477]

I kinda just went along with the ride lol, wow

[@yopxy6188]

Do it with limits

[@AnOrrangeShirt]

Would it be possible to integrate sqrt(tan(x)) twice?

[@hungryfareasternslav1823]

Yes, but that will be real pain or it is non-elementary!!!

[@lalitverma5818]

I can't imagine

[@yahyabajdadi516]

How about differentiating sqrt^3(tanx)?

[@natan9065]

Yahya Bajdadi d/dx((tanx)^(3/2)) = (2/3)(tanx)^(1/2)•(secx)^2 By the chain rule

[@maxhaibara8828]

what's the derivative/integral of N! (factorial of N) ?

[@MagicGonads]

No idea, but I found interesting that n! = int(0->+inf) (x^n e^-x dx) for x > -1 (Found that by accident) also written as int(0->+inf) (e^(n*ln(x)-x)dx)

[@maxhaibara8828]

wow he need to make a video about this

[@MagicGonads]

It's not a derivative because it has both n and x, but still suprising.

[@MagicGonads]

www.desmos.com/calculator/oukquiyntu

[@MagicGonads]

www.desmos.com/calculator/balkjdohcq The curves are calculated, not derived.

[@anthonysteinerv]

I used trigonometric substitution for both of the integrals and got the exact same inverse tanget but I got a logarithm, the thing is that I did the derivative of that anseer and I didn't get the square root of tangent x, actually I got the sine of 2x, I need help bro.

[@OonHan]

covered my ears during the time of 0:15 The integral of *I CANT HEAR ANYTHING BLA BLA BLA*

[@blackpenredpen]

lol!!! Let me integrate blah blah blah next time!

[@OonHan]

LOL

[@Wu-Li]

int of √tanx dx tanx=u² 2udu=sec²xdx dx=2udu/(u²-1) again, int of √tanx dx = int of u×2udu/(u²-1) = 2[int of u²du/(u²-1)] =2[int of (u²-1+1)du/(u²-1)] =2[int of du + int of du/(u²-1)] =2[u+int of du/(u²-1)] =2[u+½log|(u-1)/(u+1)|] +k =2u+log|(u-1)/(u+1) +k =2√tanx + log|(√tanx -1)/(√tanx +1)| +k Therefore, Int of √tanx dx = 2√tanx + log|(√tanx -1)/(tanx +1)| +k

[@upanshulakhani6221]

That's nice but why don't you check the answer by integration

[@Bludgeoned2DEATH2]

Now do it in Chinese, Dr. BlackPenRedPen!!! 😁😁

[@blackpenredpen]

Will you watch, like, and share it?

[@Bludgeoned2DEATH2]

blackpenredpen of course, just like I watched the integral of sqrt(tanx) in Chinese and liked it too!

[@blackpenredpen]

Challenge accepted!!!

[@agustindominguezcruz5000]

Now do it by the derivative definition

[@wduandy]

Love you

[@gloystar]

What a mess LOL!

[@physicsphysics1956]

Where is the mistake?

[@Metalhammer1993]

well you integrated sqrt( tan(x)) now it's time for tan (sqrt (x)) bro. let the battle begin xD

[@turbopotato4575]

Im pretty sure thats impossible.

[@Metalhammer1993]

turbo potato either impossible or even worse madness. i was only joking. i don't wanna kill him.

[@7necromancer]

integral tan(sqrt(x)) dx = i (Li_2(-e^(2 i sqrt(x))) + x + 2 i sqrt(x) log(1 + e^(2 i sqrt(x)))) + constant

[@colleen9493]

Wow there are 12^2 comments. At least until I made this one.