a magical way to solve integrals?
Рет қаралды 84,291
Күн бұрынcalculus is like magic! Visit 👉 brilliant.org/blackpenredpen/ to learn more integration techniques! (20% off with this sponsored link)
I noticed that sec(sin^-1(x) is actually the derivative of inverse sin(x) when I was teaching my precalculus class on Zoom! I was very amazed by this calculus coincident! So I investigated more on that and discovered how I could make more of this kind of "integral magic" or integral tricks by solving some simple separable differential equations. You will see two examples in the video and try to come up with similar "integral magic" on your own! This is what makes teaching fun! The more math I teach, the more ideas I have and the more mathematical connections I see.
Subscribe for more math for fun videos 👉 bit.ly/3o2fMNo
#calculus
0:00 intro
0:37 integrate the trig integral of sec(sin^-1(x))
2:01 creating your own integral magic by solving a separable differential equation
4:05 example 1
5:43 example 2
7:52 check out Brilliant
If you need more help with your class, check out my just calculus tutorial channel: / justcalculus
💪 Join our channel membership to unlock special perks,: bit.ly/34PGH4h
📸 Math notes on my Instagram: / blackpenredpen
🛍 Shop math t-shirt & hoodies: bit.ly/bprpmerch
10% off with the code "WELCOME10"
Equipment:
👉 Expo Markers (black, red, blue): amzn.to/3yYLqOf
👉 The whiteboard: amzn.to/2R38KX7
👉 Ultimate Integrals On Your Wall: teespring.com/calc-2-integral...
---------------------------------------------------------------------------------------------------
**Thanks to ALL my lovely patrons for supporting my channel and believing in what I do**
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls
---------------------------------------------------------------------------------------------------
💪 If you would also like to support this channel and have your name in the video description, then you could become my patron here / blackpenredpen
@blackpenredpen 2 жыл бұрын
7:47 Me in 2008
@hexagonist23 2 жыл бұрын
Please make a video where there is ZERO mathematics. We need to take a break from math and just relax
@crep50 2 жыл бұрын
Better idea: call this man out and show us the most complex proof you know
@donwald3436 2 жыл бұрын
So you're 26 now? lol
@sudirosudiro4218 2 жыл бұрын
kzfaq.info/get/bejne/a7ykmrmiltTSkZc.html Proof of Laplace Transform of Unit Ramp WITHOUT TABLE | ℒ{t u (t)} = 1/s^2
@createyourownfuture3840 2 жыл бұрын
@@hexagonist23 no
@reidpattis3127 2 жыл бұрын
I hate it when you try to do your math homework but accidentally prove the Riemann Hypothesis
@blackpenredpen 2 жыл бұрын
It happens.
@fasebingterfe6354 2 жыл бұрын
Same. One time I was solving my math homework when I accidentally proved the 3n+1 problem
@i-am-at 6 ай бұрын
Yeah right? I was doing my CS homework and proved the p vs np problem
@kingoreo7050 6 ай бұрын
I was just having some coffee and I solved the Navier Stokes problem
@helphowdoinputusername3571 5 ай бұрын
I woke up one morning and found that I solved all of Einstein's field equations in my sleep again. Sleep mathing is very annoying indeed.
@albertohart5334 2 жыл бұрын
Longer beard = more wisdom As t -> inf. BPRP beard -> inf.
@Shreyas_Jaiswal 2 жыл бұрын
And on-head hair tends to 0
@xarran 2 жыл бұрын
Wrong, it converges, but we don't know what value it converges to.
@gurkiratsingh7tha993 2 жыл бұрын
The hair looks a little bit disgusting but it is his choice
@kevina3673 2 жыл бұрын
Someone needs to shut this man up lmao.
@anshumanagrawal346 2 жыл бұрын
@@xarran Exactly
@logiciananimal 2 жыл бұрын
Not quite on the caliber of the discovery of noble gas chemistry by preparing a chem 101 lecture (like was done historically) but another illustration why (at least subjectively) teaching elementary courses can still be very rewarding.
@ohgosh5892 2 жыл бұрын
Every time I set about teaching some form of electro-magnetism/ physics/ electronics, I learn a little more. It's the best bit about teaching...
@donwald3436 2 жыл бұрын
Beard is also inverse of hair.
@vishalsrivastava785 2 жыл бұрын
Don't forget to watch GOW Ragnarok Trailer
@michaellarson2184 2 жыл бұрын
They really should teach you more about inverse trig functions in precalc considering how much they pop up in higher level math.
@PixelSergey 2 жыл бұрын
Assuming you didn't know this trick, would it be possible to solve the integral of tan(arcsin(exp(x)))?
@randomblueguy 2 жыл бұрын
Let u = arcsin(exp(x)) du = (1/sqrt(1-exp(2x)) • exp(x) dx We know from trigonometry that, tan(arcsin(β)) = β/sqrt(1 - β^2), which is exactly equal to du/dx when β = e^x I think many people learning calculus would immediately recognize any term resembling x/sqrt(1 - x^2) the second they see it, and substituting u = “the inside” seems to be a fairly natural move as well. I don’t think the integral would be particularly difficult if given to an average calculus student for the first time.
@PixelSergey 2 жыл бұрын
@@randomblueguy makes sense!
@anshumanagrawal346 2 жыл бұрын
I mean if I ever see trig of inv trig, first of all I will try to simplify it and then proceed, so you'll probably get a solvable integral
@somasahu1234 2 жыл бұрын
@@randomblueguy derivative of x²=2x but derivative of e^2x will be 2e^2x 🤔
@redwanburkan4790 2 жыл бұрын
You can also let e^x= sinθ (in order to get arcsin(sinθ) = θ) Therfore, e^x dx= cosθ dθ So our integral ∫ tan(arcsin(e^x)) dx Becomes like this ∫ tanθ cotθ dθ Simplifying this to ∫ dθ = θ + C And since e^x= sinθ So θ in terms of x is gonna be arcsin(e^x)
@geoffreytrang8670 2 жыл бұрын
Given any function g, we want to find a function f for which f(g(x)) is equal to g'(x). In this case, we could just define f(x) to be g'(g^-1(x)), assuming that g is one-to-one (or injective). For example, let g(x)=x^3. Then, f(x)=3(x^(1/3))^2=3x^(2/3). For g(x)=x^2, we must restrict x to nonnegative values (to avoid needing an absolute value), and f(x) would then be 2√x.
@jeffvandenbrande8455 2 жыл бұрын
So, i just finished rewatching your 100 derivatives, integrals and series vids (took me a while, but i wantend and needed a refresher). With this vid, i had a Leonardo DiCaprio- meme moment where he points to his tv screen after having a major realisation (its from “once upon a time in hollywood” Movie). This is freaking awesome!
@MeistroJB 2 жыл бұрын
Incredible as it is, it's just a magic trick, as there is no actual magic. A covert magic trick is that I can (almost) see it. Thanks, pro bro. You make it look easy.
@chessematics 2 жыл бұрын
It was pretty clear if you look at the general formula of the derivative of the inverse function. (d/dx)(f⁻¹(x)) = 1/f'(f⁻¹(x)). Let f(x) be sin(x) and magic is oncoming. This way you can generate your own "f(x)" (as described in this video) when you have your "g(x)" ready.
@tonyhaddad1394 2 жыл бұрын
Awesome idea great teacher !!!
@XAE-yc9rr 2 жыл бұрын
I'll give two more examples this integral working, besides trig functions: f(y)= e^-y where the integral becomes int(e^(- ln x)) dx f(y) = y^2 where the integral becomes int( (-1/x)^2) dx
@JohnnyLaw3134 2 жыл бұрын
That's was pure Magic!
@appybane8481 Жыл бұрын
Integral of e^-ln(-x)=-ln|-x|+c
@johnhumberstone9674 2 жыл бұрын
Just amazing!
@john1522 2 жыл бұрын
The real magic is the black pen magically coming out as blue on the whiteboard. Spooky!
@blackpenredpen 2 жыл бұрын
😆
@ididagood4335 2 жыл бұрын
Great video! I can’t tell the difference between derivatives and integrals
@andrasfogarasi5014 2 жыл бұрын
Trivial case. Integral of ln(x) is x, as the integral sign and ln cancel.
@farfa2937 2 жыл бұрын
Thanks, I'll now proceed to check if dy/f(y) = dx before wasting my time in subs/parts/whateverelse!
@Danylux 2 жыл бұрын
Help, I tried to do it with e^(2x) *sen(x) and there's no way to do the inverse or it's derivative, I tried wolframalpha and symbolab and both refused The derivative is 0.2 e^(2x) *(2senx-cosx)
@DzikaFizyka 2 жыл бұрын
That is legendary.
@donwald3436 2 жыл бұрын
When will beard reach feet?
@berzerksharma 2 жыл бұрын
Lim n->N
@official-obama 2 жыл бұрын
after he shaves it
@donwald3436 2 жыл бұрын
@@official-obama noooooooooo
@official-obama 2 жыл бұрын
@@donwald3436 most likely it will be after he shaves it
@official-obama 2 жыл бұрын
@@donwald3436 i did not get my own joke
@hatembahri4314 2 жыл бұрын
Just brilliant
@uzomannaemekateflondon1144 2 жыл бұрын
Sir what branch of mathematics did you do in your masters and PhD
@CanadianTacticz 2 жыл бұрын
Wait so can we find a general solution to that DE?
@golocok 2 жыл бұрын
Excelente "truco"
@fatho4357 2 жыл бұрын
I got a question for part f(y) = d(y)/dx Become 1/f(y) dy = dx Maybe i am wrong or maybe not, just comment if i wrong but my teacher in ODE say that That we can't separate the dy/dx, Because they are one unit, can't be separated. But maybe i wrong tq 🙏😁
@VenkataB123 2 жыл бұрын
A lotta calculus works on the fact that dy/dx can be separated. Like, we are taught that dy/dx is not "dy divided by dx" but we eventually started imagining it to be dy divided by dx to simplify stuff. It isn't wrong at all.
@kiwithemaniaguy Жыл бұрын
0:09 That jumpscared me tho Also, nice you just earned a subscriber. :)
@MohitKumar-eu4pz 2 жыл бұрын
Yeah
@28sarthakdwivedi44 2 жыл бұрын
Sir do you actually conduct zoom meetings now also??pls do share the links here also .. i m waiting..🥺👀👀😋😋
@procerpat9223 2 жыл бұрын
Teaching is the best way to learn
@_wetmath_ 2 жыл бұрын
and therefore, d/dx=sec
@gurkiratsingh7tha993 2 жыл бұрын
This is just like calculus magic, after watching this video I am sure that now everything is possible in maths, even proving riemann hypothesis😎😎
@klementhajrullaj1222 Жыл бұрын
tan(arcsine^x)=e^x/V(e^x-1) and then the integrale of this is d(e^x)/V(e^x-1)=d(t)/V(t-1)=d(t-1)/V(t-1)=d(u)/Vu=2Vu+c=2V(t-1)+c=2(e^x-1)+c. 😀😉
@raymondgreen1627 2 жыл бұрын
Hi sir can you teach derivatives of the inverse of a trigonometric functions and differential calculus
@user-wu8yq1rb9t 2 жыл бұрын
Just WOoOW Great 🤓
@Jamesdaman69 Жыл бұрын
If you put in ln(x) you get the inverse of the logarithmic integral
@ZipplyZane 2 жыл бұрын
I couldn't help but read the second line on the right as int f(goo) dx = g(x).
@aronpacino8009 2 жыл бұрын
i can't unsee it now 😐
@Hei_Darkfire 2 жыл бұрын
I am so glad my Calculus teachers haven't tried to screw me up with this one. If they had, I absolutely would have gotten it wrong.
@TruthOfZ0 2 жыл бұрын
wow wait can you do that also with a sum and not only integral???
@angelodecastro_ 2 жыл бұрын
The trick’s kinda crispy tho 👌
@jesusdamianrodriguezpinto7514 2 жыл бұрын
Another magic integral would be 1^x. It's short but interesting :D
@gileadedetogni9054 12 күн бұрын
I liked that man hahaha
@kruksog 2 жыл бұрын
This vid needs a Dr Peyam "wauwwww." I haven't been this shocked by a piece of calculus in years... Since I took analysis really.
@blackpenredpen 2 жыл бұрын
Glad u liked it!
@Math342010 2 жыл бұрын
Interesting.....
@petereziagor4604 2 жыл бұрын
Really awesome
@blackpenredpen 2 жыл бұрын
Thanks.
@petereziagor4604 2 жыл бұрын
@@blackpenredpen ahaaa, go live I'm awake already 😁 4: 22 am here
@sabarik6698 2 жыл бұрын
Interesting...😲
@kepler4192 2 жыл бұрын
now integrate csc(arcos(-x))
@TheMemesofDestruction 2 жыл бұрын
“Do you believe in magic? In a Calc Teachers pen?” 🖊 ^.^
@tonyhaddad1394 2 жыл бұрын
At first i thought that you know d(integrale(f(x))/dx = f(x) also 😁
@noahtaul 2 жыл бұрын
Uh… wouldn’t it be easier to choose g first, and then let f(x)=1/(g^-1)’? That way there’s no annoying integral.
@aronpacino8009 2 жыл бұрын
why do you have to take the inverse of g, tho?
@SpeedyMemes 2 жыл бұрын
nice!
@user-wu8yq1rb9t 2 жыл бұрын
Now I'm here! Because I want to check which one is better! The wrong way or the correct way?!
@emirhalilovic3362 Жыл бұрын
integral of (-1/x)², cancel the integral and square and get -1/x ⛔ edit: I almost forgot... +C
@vladusa 2 жыл бұрын
Not all integrals wear d/dxs. Some just cancel.
@Mehak_jindal__ 2 жыл бұрын
U niced it
@aayushkadiwar7496 2 жыл бұрын
I LOVE UR VIDS BPRP
@sambhusharma1436 2 жыл бұрын
You are awesome
@frozenkingfrozenking6989 2 жыл бұрын
Good
@samsam4455 2 жыл бұрын
Dear blackpenredpen, I would appreciate if you can check the convergence of the séries: ((ln n)^n)/n! And (n^ln n)/n!. Many thanks Sir
@NonTwinBrothers 2 жыл бұрын
Oh believe me, he can teach more than just Pre-Calc
@mrsanjaythakur57 2 жыл бұрын
Thsnk you sir
@indarajgochermaths5176 2 жыл бұрын
nice
@iAzazelHD Ай бұрын
And how would you actually integrate tan(sin^-1(e^x))? It scares me too >///
@ichvvhovxugcvuhi3522 2 жыл бұрын
Express beard as a function and double integral find volume of beard.
@JM-us3fr 2 жыл бұрын
Your beard is looking amazing bro
@blackpenredpen 2 жыл бұрын
I appreciate your comment : )
@buxeessingh2571 2 жыл бұрын
The original integral is hilarious.
@ugojlachapelle 2 жыл бұрын
That last one is so evil.
@cxpKSip 2 жыл бұрын
Huh.
@bornoflightningthunder8162 2 жыл бұрын
Damn
@AMANKUMAR-zb3zv 2 жыл бұрын
Blackpenredpenbluepen
@jacobjac3835 2 жыл бұрын
I like your hair, please give it back QAQ(BUT I ALSO LIKE THE WAY YOU ARE RIGHT NOW)
@user-fm9vk6ws6e 2 жыл бұрын
物理的數學
@dwightkurtschrute2551 2 жыл бұрын
blue pen red pen
@thephysicistcuber175 2 жыл бұрын
Hold up wtf, it's literally arcsin x.
@itsME-dc4vm 2 жыл бұрын
nice ;D
@abhirupbasu9298 2 жыл бұрын
0/0=8 ; how i can solve it ?
@vinayakm7800 2 жыл бұрын
മലയാളീസ് undoooi...
@sudoheckbegula 2 жыл бұрын
Why does he look like Dalai Lama
@frankcabanski9409 2 жыл бұрын
When bad teacher with marbles in his mouth and a silly, gimmick mic takes 8 minutes to teach something that could take 1 minute to teach.
@blackpenredpen 2 жыл бұрын
Hi Frank. How are you? Hope you are doing okay.
@User-he6zd 2 жыл бұрын
Lmao do you have nothing else to do you're like a 50 year old man still commenting irrelevant mean spirited comments on all of this guys videos, which are aimed at a younger audience to educate will little prerequisite knowledge. Just makes you come off as weird and sad, esp. as you've made so many comments like this, it's pathetic
@frankcabanski9409 2 жыл бұрын
@@User-he6zd Ditch the silly mic. With my pencils and math knowledge tied behind my back, I can out math this guy, I am a tutor. This guy uses a gimmick and takes the long way to explain things.
@Videogamewrestling22 2 жыл бұрын
@@frankcabanski9409 Jesus christ.... lol he's holding a mic in his hand so his viewer's/students can hear him better. 🤣 your obnoxious for a guy that teaches fucking basic math you can learn at a library for a $1.50
@Videogamewrestling22 2 жыл бұрын
@@frankcabanski9409 also your videos take way longer
@adamnevraumont4027 2 жыл бұрын
This magic trick looks like a failure of notation. sin^-1 is the inverse of the antiderivative of the reciprocal of sec. So A f(AR(f)^-1) = AR(f)^-1. Differentiate and get f(AR(f)^-1)=D(AR(f)^-1). Which starts looking like algebra, not calculus.
@dlevi67 2 жыл бұрын
Please stop indicating inverse trigonometric functions as "fˉ¹(x)". It is extremely confusing and just plain wrong.
@ethohalfslab 2 жыл бұрын
It isn't wrong, though. It's accepted notation.
@dlevi67 2 жыл бұрын
@@ethohalfslab So are a lot of other things; it doesn't make them any less confusing or wrong. Particularly if you want then to indicate f(x)·f(x) as f²(x).
@ethohalfslab 2 жыл бұрын
@@dlevi67 are you gonna go yell at a 3rd grader "USING X FOR MULTIPLICATION IS WRONG"? I wouldn't think so, because that notation is fine for their usecase. If this math problem used something like f^2(x) for something, then it's fine to request arcsin and the like. But it doesn't use f^2(x).
@dlevi67 2 жыл бұрын
@@ethohalfslab Except that: 1. BPRP uses that notation consistently - just as he uses consistently f²(x) instead of the longer (f(x))². 2. The multiplication symbol is _not_ an 'x'; it's '×' - it's a different notation and not an abuse. 3. If you are teaching, you should teach best practice compatible with your (average) student's ability to understand and knowledge. Someone who follows a video like this one _does_ understand and knows what f²(x) means even if it is not used in this specific case.
@ethohalfslab 2 жыл бұрын
@@dlevi67 well if you care about best practice so much, best practice is to write 1/f(x).
@mtaur4113 2 жыл бұрын
When you teach math for 10 years, the hair all moves to the other side of your head.