Area between ln(x) and (ln(x))^2 | Dear Jeff
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- Опубликовано: 8 янв 2019
- Dear Jeff, Area between ln(x) and (ln(x))^2, ft. DI method
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So 1=e in China.
So that's the secret behind chinese economic boom, calculating with 1 is WAY easier than calculating with 3!
Sensei LOL
@@sensei9767 lol
@@sensei9767 2.7
I laugh too hard in this
I thought integrating lnx-(lnx)^2 will be painful but he solved it with an ease. Great !
Could say he solved it with e’s
@@BlokenArrow Yeah! the power of exponentials imao
I did it that way, and can confirm it was painful (lots of product rules, Chen Lu, and even a quotient rule). I tried using the DI method for the first time and it is so much easier than the usual verbose "integral v du = vu - integral u dv" stuff, why wasn't I taught that 15 years ago?!
The integral of (ln x)^2 turns out to be x((ln x)^2 + 2 - 2 ln x)
almightyhydra Because the education in the West for mathematics is bad
Angel Mendez-Rivera, not just the west bud
2:09 when you make a joke but realize no one actually laughed
I actually found it interesting.... But yeah I get what you mean and I agree xD he's a nice guy tho
LMAO
😂
I laughed so ur wrong
Yeah, phonemes correspondances in different languages differ. I cannot force people to laugh at this. After all, we're not African Americans.XD
Holy cow, this is the first problem that I actually managed to solve by myself (I'm in high school and watch your videos just for fun). Also I've managed to get full points on my latest graduation preparation exam in maths and I genuinely think your videos are one of the reasons! Thanks so much ;)
Inferno Captures wow! That's amazing. Thank you!
Hey bro same here
@@blackpenredpen I'm in eleventh grade (in france) rn and you made me love math i still don't understand how you do integration tho, i know things like u substitution or even the base principle of integration (how it works) but i don't know how to do it.... last time i managed to find the complex solution of an equation with exponentials and natural logs and i was so proud! Thank you for everything bprp ;)
@@alexandrebriard9175 T'es en terminale? Regarde les videos de Eddie Woo il fait plus des "cours" que des exemples compliqués comme ici. Sinon ya aussi ivan monka mais jpense que tu connais
Same
5:48 This is how King Crimson works
I can't escape the JoJo references anywhere on RUclips 😭🤣
even now jokes about JoJo are funny lol
loool
But 3=e so 3-e=0 isn't it?
3-e is about 0.28 because e is about 2.7182~
@@no_mnom whooooooosh
Doesn't e=2 and pi=3?
**Isn't it**
@@dekrain And i = !
So e^(pi*i) = 2^(3!) = 64
The english “e“ sounds the same as the german “i” so every time somebody uses “e” in an english math video I always first think: “why is he/she using complex numbers there?”
Same in Italian
Same in spanish
The same happens in portuguese.
Same in french
Every European: says i as /i/
English user: says i as /ai/ and e as /i/
I absolutely love your Math videos!!!!
Please continue with what you're doing ❤️
Sending you lots of love.
The sound becomes out of sync by the end.
Joshua Hillerup
Oh man, I didn't realize it. I noticed there was a weird jump around 6 min then the video became weird.
according to engineers, there's no area between the functions
When you have studied calcus, completed your engineering masters and then get mind blown by the D/I method for integrating by parts!!!
That u-sub was the sexiest thing I've ever seen
Very clever problem👍🏻 nice use of tabular parts!
I haven't take calculus yet, but I like to watch your videos even if I don't understant everything. Thank you!
Another approach to evaluating this integral is to factor out an ln(x) term from ln(x) - (ln x)² to get ln(x) - (ln x)² = ln(x)(1 - ln x), and then use integration by parts with u = 1 - ln(x) and dv = ln(x)dx (the u and dv notations are heuristics, intended to denote which function should be differentiated and which should be integrated; in this case, you differentiate 1 - ln(x) and integrate ln(x)).
This is a sweet answer. I like this problem. Thanks for sharing.
You can simply integrate ln(x)^2 using the DI method! (The integral of ln(x) is xln(x)-x)
Wow you managed to solve this difficult problem with e's
Hahaha thanks!!
integration-by- parts............ woooow that was so simple trick to sove it
thanks
Thank you
To an engineer, the answer would be 0
@@cloviselguedino4708 you mean Physicist right?
@@cloviselguedino4708 yes those physicians using calculus.
so... who else needed this video because their college homework was too hard for them :/
Excellent explanation
Pls make videos very fast as you did in the past. Because I love to watch your videos
0:40
blackpenredpen is cowboy confirmed?
And as always a very good video.
400k subs by the end of this year? Man, you're optimistic with math lovers 😅
thanks
I sat down & did it for myself.
Damn I feel smart...
Just separate the two terms and solve it individually, lnx will be xlnx-x and integrate the other term using by parts method wherein you can write it as 1*(lnx)^2. I find this way much easier.
1 = e as an approximation works sometime, especially for order of mangnitude estimates.
Neat, sweet, complete!
Fred
I don't like maths and I don't understand anything about this except for a few things, but lately I've been watching anyway a lot of videos of yours lol, really interesting
Cencio your brain must be burning
@@sonpham3438 actually no, I love to see how far we've arrived with maths, calculating the factorial of a non natural number, the fact that we can actually calculate the area of a function, the poincare objection and so on, that's crazy!
Cencio oh I see
@@fraaaancesco Are you sure you don't like maths? Because it sounds like you really like maths.
This men is a genius
Nice video!!! I see you from Argentina.
I think therere is no need to substitute, it's quite easy to integer by parts, for example f' = 1 and g=log x per int log x and f' = x and g = log^2 (x) for int log^2 (x)
Use DI method to integral of ln(x)^2 = x*ln(x)^2 - integral of 2ln(x). and integral of ln(x) is xln(x) - x. So, 3(xln(x) - x) - x*ln(x)^2 from 1 to e is the answer.
cheers, jeff
My respect
so thats great!
(ln x)^2 is my favorite trick question on exams! I can't tell you how many students will try 2 ln x.
I don't remember asking for this... but I like it :D
I'll be doing engineering math 2 this year, so his will help anyway hehehe
Jeff : )))))
Maneme ejeff
2:08 that escalated quickly
If you know that an antiderivative of ln x is x ln x - x, it's easy to guess that an antiderivative of ln^2 involves x ln^2 x. A little work reveals x ln^2 x - 2x ln x + 2x as an answer, so we just evaluate (x ln x - x - x ln^2 x + 2x ln x - 2x) at e and 1 and subtract the first from the second. Your way works too, I guess. :)
That’s a cool lookin watch bprp!
Oscar Troncoso thanks!!
一 means 1 in Chinese and e in English! Love it!
2:03
to comes after e.
You are an integration wizard and inspired me to practice more again, so I can also become one ^^
Ok so, you can now calculate the angle between two curves at (1;e) point.
I got 45° for the intersection at (1, 0) & ~16.25° at (e, 1).
my name is jeff, and i am a math major
thanks for the gift
Jeffery Sahu yay!!!
كان من الأفضل الاهتمام بحساب الحد الأقصى للقطعة العمودية التي تتحرك داخل المساحة (بين 1 و e)!
it was more interesting to calculate the maximum of a vertical segment moving inside your surface (between 1 and e)!
Or the maximum of a horizontal segment moving inside the surface !
dude you are hero 13 students of geology are in ur debt because of this video .this thing was in final
You can also easily find the inverse functions of these both and find the area under y axis, which gives the exact same result as the u sub you used.
Manan Seth The function is not invertible.
@@angelmendez-rivera351 Explain please?
Manan Seth What do you want me to explain? The function f(x) = log(x) - log(x)^2 is not invertible, so inverting the function and finding the answer integrating with respect to y cannot be done. Even if it could be done, though, it would be significantly more complicated and an unnecessary amount of work. f(x) being not invertible is a fact, so I am failing to understand what about it needs explanation. Why is it not invertible? There is no why, it just is not invertible.
@@angelmendez-rivera351 No you can invert y = ln x (x = e^y) and y = (ln x) ² and then find area between the two curves because it'll remain the same. Another way of saying is you can integrate the same graph in the form of integral x dy instead of integral y dx.
Woohoo! I did it before watching the video! (Aside from getting the problem, obviously). It may be easy for most of you but I’m just learning Calculus and teaching myself via YT vids and google searches. Wow. I’m proud of that one. 3 - e all day baby.
Jaxon Holden NICE
just for a fun fact, e sound is 2 in Korea XD
So,
e = 1 in Chinese
e = 2 in Korean
e = 3 in Spanish (see Andres Osorio Santamaria
's comment)
isn't 2 'dul' in Korean? I strictly remember the counting as 'hana', 'dul', 'se' or 'sae' and so on... am I wrong?
@@Uni-Coder 3 is "tres" in Spanish
Guillo 15 You missed the joke. e = 3 is a joke about engineering.
@@Kitulous korean has two different number systems. first system is 'hana', 'dul', 'set', 'net', so on. And second system is 'il', 'i(e sound)', 'sam', 'sa', so on. we use second system when we do mathematics.
man that's so cool
I actually managed to do it myself. And then of course my trust issues always make me watch the video to check the answer
I managed to do this quite easily, which I'm proud of.
I have a very interesting limit: as you may know f(x) = (1+1/x)^x has limit e in infinity. What is Lim x -> Infinity of 1/(1-f(x)/e) -2x. I could't work it out but Wolfram gave me a value i did not expect at all
Hey i hv an idea how abt some videos on probability, pnc and game theory
magical e.
Can found integral lnx= -1 and integral lnx^2=e-2 then area=abs(-1)-(e-2)=3-e
Please help me to find area of curve r= sqrt(3)*cos(3*theta)+sin(3*theta). I am not find answers pi/3.
Nice proof of e < 3 if your definition of e is the zero of log(x) - 1, with log(x) being an anti-derivative of 1/x.
A similar proof is to show that the integral of log(x)^3 from 1 to e is 6 - 2e, also via integration by parts.
On the other hand, if you use the series definition e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ..., then you can prove that e < 3 with series comparison: e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ... < 1 + 1/2^0 + 1/2^1 + 1/2^2 + 1/2^3 + ... = 3.
I thought you would integrate in y.
Int int f(x,y) dx dy.
Which would lead to exp(y) and exp(sqrt(y)).
Altogether, nice solution.
2:02 so if the integral goes from 1 to e, it’s... just 0, right?
Nice video.
You could have factored out the e^u and it would be much easier😂😂😂
Sometime we make the problems much harder by accident.
Also I have a question :
Could we calculate it by a double integral? (I mean the one with a D in the bottom of the double integral).
We just learned about it yesterday and I wanted to know if it can be solved by it.
I have *never* seen that integration by parts technique before
AstroTibs check out DI method on my channel.
@@blackpenredpen You're on.
I didn't see yet volume integral related here... but maybe someday.
Black pen GREEN PEN
bprp - "dirty work thing....integration"..
me- "This man is a god...
tries, tries and tries,...
subscribes :)
My TI-36X Pro returned the same numerical answer, 3-e.
/Lonewolf
5:50 magics of mathematics yee
the audio got unsynced during the middle of the video lol
One integration by parts done. An infinite number more still to go. Life doesn't get much better thab this until you discover number theory.
Can you please find the area between y = x^(1/(1+x)) and y = x^(1/(1-x)) ?
How did u draw the graph of the later function......I mean how would I know what to draw in a test......
It's obvious
You would not need to draw it on a test. However, you first want to find any asymptotes the graph has. You also want to find critical points, the points of inflection, and behavior towards the infinities. You also want to know the concavities at different intervals.
I factored out ln(x) and integrated by parts…a lot messier but same answer
Hey BPRP - wanna try
int sqrt(log(x)) dx
? Goes with int sqrt(tan(x)) dx, int sqrt(exp(x)) dx, and the like in terms of the general pattern, and offers a not-instantly-obvious appearance of the imaginary error function which you introduced some videos back.
Ok, will do!
@@blackpenredpen zomg! Thanks!
Mister blackpenredpen I want you to solve this problem calculate the remainder of the division of (cos(a)+Xsin(a))^n by X^2+1
Sir where can i get more beautiful under graduate math problems? Plz tell me so
Yet another proof that e
What about the area between zero and 1 bound by both curves :)
pi = e = 1. Nice
what
It would be cool to calculate the area between a hyperbolic and a ellipse
Many videos on it search it up
I solved it in my mind.
Given the thought that he typically uses black and red … yet he also uses blue and green. Mmhmm.
おー
これは「小テストにでるぞー」て感じの問題ですね 10点もろたで😄
How do i ask for some math problems to be solved by you sir? I was not able to find your email or anything.
I realized when you changed the bounds of the integration it becaome pretty much my strategy, which was to rotate the plane and do int(e^u-e^√u) from 0 to 1
Fractal No, that is not the same as what he did. The function he had was different from yours. I’m not even certain your integral gives the correct answer, as I am evaluating it right now.
@@angelmendez-rivera351 actually it is e^√u - e^u
but it is the same thing
I have done it
and it also is just the inverses in a proper defined integral
including the bounds.
@@angelmendez-rivera351 Im saying that when he substitutes he gets a part of the integral I would get anyway. obviously Im not saying he has done the exact same thing I did.
afterall the substitution is literally the same process as rotating the plane, even though its just a part of it. being unnecessarily pedantic in an analogy as simple as that is foolish
Fractal I’m not being pedantic, I’m literally just correcting what you said. Your original comment says it was the same strategy. I explained it wasn’t. If correcting something inaccurate is pedantic, then you must really hate professors.
Anyone else got a problem with the video being faster than the audio.
Yes I did.
Yes I noticed that 🤔
Me too.
I actually had the inverse problem (problem¯¹), the audio was faster than the video
Not me
Teorem of stokes please, thanks Friend
yay i got it righttt
I have solved it in less than one minute
I have a question about how to solve : f'(x)=f(x+1)
How to calculate arc length of sinx from 0 to pi ?. I stuck at the integral( sqrt of (1+(cosx)^2) )dx
anik bhowmick I’m not completely certain, but I think the integral is not doable without using elliptic integral functions. So, no closed form.
Mayve try to use the trig Pythagoras sin^2(x) + cos^2(x) = 1
or maybe use hyperbolic trig identities
thomasweis Weis That does not help. All it does is give him [sin(x)]^2 + 2[cos(x)]^2 under the radical. sin(x)^2 = 1 - cos(x)^2, not 1 + cos(x)^2. Hyperbolic functions do not help either because introducing them requires a change of variables and the use of chain rule, which turns the integrand into another expression whose elementary anti-derivative does not exist.
The answer to your integral is SqRt(2)·E(π|1/2), where E(x|k) is the incomplete elliptic integral of the second kind with trigonometric coefficient 1/2. This is the only way to express the answer. There is no possible way to write using elementary functions such e^x or anything of the sort.
In fact, generally speaking, you cannot calculate the arc length of trigonometric functions in closed-form.
If e is 1 in china , how do the chinese name the number of euler ?
4:23 what is that? Can anybody tell me how it works?
Integration by parts.
@@quocanhnguyenle4952 Thì biết là nguyên hàm từng phần nhưng cái cách viết thành dòng kia xong cộng trừ luân phiên các thứ. Cái cách làm đấy từ đâu ra?
Great video! Just a (possibly stupid) question, but here goes:
Why isn't the area between lnx and (lnx)^2 considered from -inf to 1 as well? As in:
Total area = integral of (lnx + (lnx)^2) from -inf (edit: 0) to 1 + integral of (lnx - (lnx)^2) from 1 to e.
Would really appreciate a response.
Thanks!
Edit: I meant 0 to 1. The value at which lnx becomes -inf, sorry!
because the area between the bounds of 1 and e is the only finite area between the curves.
area -inf to 1 and e to +inf are both just +inf
@@zackologlu7018 hey, thanks for the reply. I added a weird reply here and didn't check that the question was worded with wrong limits. I've edited it, thanks!
Sarthak Varshney Fixing the limits does not change his answer to your question. The integral from 0 to 1 diverges automatically, and this because both Ln(0) and [Ln(0)]^2 are infinitely large. It literally only makes sense to talk about the area between these two functions in the interval [1, e]. No other interval makes sense to consider.
@@angelmendez-rivera351 Hey, thanks. The reason why I'm asking this is because integral of lnx from 0 to 1 is -1 and and (lnx)^2 is 2.
I agree that ln(0) is -inf but the integral is xlnx - x and when x->0, xlnx becomes 0.
Hope that helps in showing my point.
Sarthak Varshney In mathematics, the terminology “the area between two curves” does not mean what your intuition interprets this phrase to be. Instead of imagining the phrase to be in English, think of it as being written in some foreign strange constructed language that uses poor nomenclature to describe something. What that phrase REALLY means in this case is literally just the area from 1 to e. That is by true by definition. It has no explanation other than that is what the phrase was chosen to mean. In general, if you have two functions f(x) and g(x) and you are asked to find the area bounded by f(x) and g(x), what they are asking you in complete precision is J1 + J2 + ••• + J(n), where J1 represents the integral of f(x) - g(x) from a1 to b1, where a1 and b1 are the endpoints of the left-most interval (a1, b1) such that f(x) - g(x) is continuous in this interval and f(x) - g(x) is bounded in this interval, J2 is the integral of f(x) - g(x) from a2 to b2, where a2 and b2 are the endpoints of the 2nd left-most interval with the properties already listed, etc. and J(n) represents the same integral over the same difference from a(n) to b(n) where a(n) and b(n) are the endpoints of the nth left most interval with the listed properties. If the number of these such intervals is finite, then we get a finite series, and the nth left most interval with these properties is also the right-most interval with those properties. That is just the mathematical definition of the phrase, which contradicts the meaning that the phrase has if you interpret the words as you would in English. There is no explanation why that can answer your question because that is just the definition which was chosen. If you want to ask why this definition, best not bother, because you are not going to find this answer.
5:53 to end: off-sync audio/video