a magical way to solve integrals?
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- Опубликовано: 9 июл 2024
- calculus 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.
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#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
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7:47 Me in 2008
Please make a video where there is ZERO mathematics. We need to take a break from math and just relax
Better idea: call this man out and show us the most complex proof you know
So you're 26 now? lol
ruclips.net/video/5VtgVq1nnac/видео.html
Proof of Laplace Transform of Unit Ramp WITHOUT TABLE | ℒ{t u (t)} = 1/s^2
@@hexagonist23 no
I hate it when you try to do your math homework but accidentally prove the Riemann Hypothesis
It happens.
Same. One time I was solving my math homework when I accidentally proved the 3n+1 problem
Yeah right? I was doing my CS homework and proved the p vs np problem
I was just having some coffee and I solved the Navier Stokes problem
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.
Longer beard = more wisdom
As t -> inf. BPRP beard -> inf.
And on-head hair tends to 0
Wrong, it converges, but we don't know what value it converges to.
The hair looks a little bit disgusting but it is his choice
Someone needs to shut this man up lmao.
@@xarran Exactly
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.
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...
Beard is also inverse of hair.
Don't forget to watch GOW Ragnarok Trailer
They really should teach you more about inverse trig functions in precalc considering how much they pop up in higher level math.
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.
Assuming you didn't know this trick, would it be possible to solve the integral of tan(arcsin(exp(x)))?
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.
@@randomblueguy makes sense!
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
@@randomblueguy derivative of x²=2x but derivative of e^2x will be 2e^2x 🤔
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)
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!
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.
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.
Awesome idea great teacher !!!
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
That's was pure Magic!
Integral of e^-ln(-x)=-ln|-x|+c
Just amazing!
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)
The real magic is the black pen magically coming out as blue on the whiteboard. Spooky!
😆
Trivial case. Integral of ln(x) is x, as the integral sign and ln cancel.
Great video! I can’t tell the difference between derivatives and integrals
Sir what branch of mathematics did you do in your masters and PhD
Thanks, I'll now proceed to check if dy/f(y) = dx before wasting my time in subs/parts/whateverelse!
That is legendary.
Wait so can we find a general solution to that DE?
0:09 That jumpscared me tho
Also, nice you just earned a subscriber. :)
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.
Glad u liked it!
Hi sir can you teach derivatives of the inverse of a trigonometric functions and differential calculus
This is just like calculus magic, after watching this video I am sure that now everything is possible in maths, even proving riemann hypothesis😎😎
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.
Just brilliant
Just WOoOW
Great 🤓
Sir do you actually conduct zoom meetings now also??pls do share the links here also
.. i m waiting..🥺👀👀😋😋
If you put in ln(x) you get the inverse of the logarithmic integral
Excelente "truco"
Another magic integral would be 1^x. It's short but interesting :D
I liked that man hahaha
Teaching is the best way to learn
The trick’s kinda crispy tho 👌
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. 😀😉
wow wait can you do that also with a sum and not only integral???
Yeah
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 🙏😁
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.
“Do you believe in magic? In a Calc Teachers pen?” 🖊 ^.^
I couldn't help but read the second line on the right as int f(goo) dx = g(x).
i can't unsee it now 😐
and therefore, d/dx=sec
At first i thought that you know
d(integrale(f(x))/dx = f(x) also
😁
nice!
Now I'm here! Because I want to check which one is better! The wrong way or the correct way?!
Interesting...😲
Interesting.....
Really awesome
Thanks.
@@blackpenredpen ahaaa, go live I'm awake already 😁
4: 22 am here
I LOVE UR VIDS BPRP
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.
why do you have to take the inverse of g, tho?
now integrate csc(arcos(-x))
You are awesome
Not all integrals wear d/dxs. Some just cancel.
integral of (-1/x)², cancel the integral and square and get -1/x ⛔
edit: I almost forgot... +C
When will beard reach feet?
Lim n->N
after he shaves it
@@official-obama noooooooooo
@@donwald3436 most likely it will be after he shaves it
@@donwald3436 i did not get my own joke
Oh believe me, he can teach more than just Pre-Calc
Good
nice
U niced it
Thsnk you sir
And how would you actually integrate tan(sin^-1(e^x))? It scares me too >///
Damn
nice ;D
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
Express beard as a function and double integral find volume of beard.
Your beard is looking amazing bro
I appreciate your comment : )
That last one is so evil.
I like your hair, please give it back QAQ(BUT I ALSO LIKE THE WAY YOU ARE RIGHT NOW)
Blackpenredpenbluepen
The original integral is hilarious.
物理的數學
blue pen red pen
Huh.
Hold up wtf, it's literally arcsin x.
0/0=8 ; how i can solve it ?
മലയാളീസ് undoooi...
Why does he look like Dalai Lama
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.
Hi Frank. How are you? Hope you are doing okay.
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
@@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.
@@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
@@frankcabanski9409 also your videos take way longer
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.
Please stop indicating inverse trigonometric functions as "fˉ¹(x)". It is extremely confusing and just plain wrong.
It isn't wrong, though. It's accepted notation.
@@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).
@@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).
@@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.
@@dlevi67 well if you care about best practice so much, best practice is to write 1/f(x).
When you teach math for 10 years, the hair all moves to the other side of your head.