integral of ln(x) from 0 to 1
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- Опубликовано: 30 ноя 2024
- improper integral of ln(x) from 0 to 1, two ways,
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when oon han said "we'll be using inverses" his method hit me immediately
Yeah, brilliant ideas are so obvious when you know them, and so hard to figure out when you don't :q
@@bonbonpony this is a really great and underrated comment
When I saw him talking about the area I immediately think about using inverse 🤔
tfw you're trying to find the mass of a uniformly dense sphere using triple integrals
Same
Four years of Engineering and I'm being out-mathed by a youngster. Lol. Well done.
He ain't even a youngster he's a developed sperm ffs , he is supposed to be doing 1+2 :)))
@@danibaba7058 someone is salty
@@danibaba7058 Did someone take a piss on your integral?
@@t0k4m4k7 this made me lol
@@t0k4m4k7 rofl
"I wanna be a cat"
Best line ever
Deepto Chatterjee *your comment is loved by Oon Han*
Welp, one girl wanted to be a Mermaid, and guess who's she now as an adult? :J
She works as a Mermaid in Disneyland ;)
@@bonbonpony Are you the same Bon Bon who watched Hobo Ryan's MLP reaction?
@@assassin01620 Yes. Why?
@@bonbonpony didnt expect to find you on a math channel xD
In case anyone was wondering, the visual intuition behind integration by parts is essentially the "flipping" observation made by Oon Han. That's why parts is particularly useful when dealing with inverse functions like arctan(x)
Thank you, IBP always seemed like voodoo, now it makes more sense to me.
Damn the second one was brilliant
Thanks!!
What about doing a video about the improper integral from 0 to +inf of x^n/(e^x-1) with n>=1 and natural?
@@blackpenredpen not due to you!
Damn, how did you draw that first integral symbol so neatly?
Mikhail Asimov
He used crayon to make whiteboard whiter
damn
More like "Damn! Why didn't I think of that???"
michael bay fixed it in post
another way to do it is by making a substitution x=e^u, which makes dx=e^u du. changing the bounds, you get the integral from 1 to e of ue^u du and solve using integration by parts
Yes,im using that too,but nonetheless,both brilliant in their own way.
Well you still technically used integration by parts equivalently to integrating lnx
Omg he is so creative to come up with that solution
Watching talented kids smarter than me makes me feel depressed
some people are just luckier than others.
DerToasti It's got nothing to do with luck/chance
It's about nurture and (possibly) talent
which a little kid like that can't influence, so pure luck.
Don't be. I'm in general happy that the world has people in it that can do things I can't, I just have to hope that they use their powers for good.
Agreed. Pure luck.
I thought about 'flipping' or rotating functions when I took Cal 2, but my professor said that one can't just claim to 'rotate' the integral to make things easier, what one ought to do is multiply it by a rotation matrix (which is just the rigorous way of 'flipping' or 'rotating.')
So Cal 2+ students, just use this: en.wikipedia.org/wiki/Rotation_(mathematics)#Two_dimensions
You don't need to know Linear Algebra or anything, and this will let you be totally correct in your reasoning!
When you rotate by 90 degrees, you will only be multiplying by 1, -1, or 0, and your result will look like Oon Han's!
Not necessarily. You can as well reorder the equation algebraically ;) E.g. when you have:
y = ln(x)
you can raise the base `e` to both sides to get:
e^y = e^ln(x)
and the logarithm will cancel on the right-hand side:
e^y = x
so now you have a function `x(y) = e^y` which you can integrate with respect to `y`:
ʃe^y·dy
from `-∞` to `0`, as in Oon Han's video :>
That's true. I originally thought of rotating only because I didn't want to integrate with respect to y, for some reason I thought that was silly and I wanted to 'fix' it so I could integrate wrt x. Particularly for integrating using the disk method, as it just made more sense for me if the disks went around the x-axis, lol.
Why? I'm integrating a different function that just happens to be the same area. It's not substitution.
This makes me wonder what other functions could be integrated easier after mirroring it.
Of the elementary functions: all the inverse trig functions and their hyperbolic cousins.
Do not think of it as mirroring, but rather as integrating along a different axis.
So for y=ln x we are integrating with respect to dy
As long as you choose the limits of integration wisely: pretty much every function ;)
You just need to make sure that after flipping around it won't be multi-valued ;)
Sqrt(x) comes to mind. Like gyroninja said, consider x=y^2.
@@gyroninjamodder That's a really good point and indeed a better perspective imo.
Oon Han is a beast for his age
*is a beast
John Days huh????
@John Days why are you on an educational video taking religious stuff? not everybody believes in your religion you know?
@John Days why?
@John Days how do you know that your religion is true in the first place
Dude when a saw a kid a was surprised , in his age , i was still peeing in my bed
. . . . . .
I'll leave it up to others
I'm smiling so hard right now, you don't even know
Alexander Roderick yay!!!
It's magic, that kid know higher mathematics! Wonderful
Those kids are adorable. Great way to rethink the problem.
Very nice. In fact carefully setting y = lnx, and therefore x = e^y makes it easier. As shown in the video, it is important to at set or identify the correct bounds. So, if x-->0, y--> minus infty, and x-->1, y--> 0, so the bounds are from -infty to 0. The trick here is to know that the integral of lnx is minus of the exponential, before evaluating. Thanks.
I had actually considered the 2nd as a method but didn’t know if I was right until I was amazed he (Oon Han) had also found it! He’s incredible for his age!
"Believe in your limits!" - love it.
Trick:
It is just the rotated + mirrored area of eˣ, just negative, so it is the negative integral of eˣ from -∞ to 0. That is a much easier function to integrate.
My favorite way to evaluate this integral is using Feynmans method. You can define a function I(t) equal to int integral between zero and 1 of x^t dx which is an integral you can evaluate to find (x^(t+1))/(t+1) between zero and one or just 1/(t+1) Then you can find I’(t) by differentiating the original integral I(t) and distributing the derivative into the integral as a partial derivative to find that I’(t) is equal to the natural log of x with respect to x then you can set the two equal
Meanwhile my classmates don't know how to set up a table for a hyperbola
My classmates don't even know what hyberbola is
I don't know what the hospital rule is but the limit of xlnx when x tends to 0 is equivalent to -ln(X)/X when X tends to + infinity with X=1/x and this limit is 0 by comparative growth.
L'Hôpital's Rule is exactly comparing growth rate. The basic idea is that you have two functions: f and g, and the quotient f/g is indeterminate (0/0 or ∞/∞). The limit of f/g equals the limit of f'/g', where f' is the derivative of f and g' is the derivative of g. Basically what L'Hôpital's Rule does is check to see which function is growing more quickly, and from there the limit can be inferred by that how much bigger one is than the other. If f'/g' is still indeterminate, then you can do it again to check f"/g", and if that is indeterminate still ... you get the idea. There is no limit (heh) to how many times you can apply L'Hôpital's Rule to a specific problem.
Awesome Job Oon Han! I've seen this method before, but did not even think to use it!
Just subscribed to your channel...
Shared it with my financial engineering team.
Great content, great image.
Keep up with the good work!
Thank you!!!
Incredibly badass solution to use the symmetry of inverse functions! Very impressive!
about 6:00 the limit could solve using the remarcable averge
lim xn->0 of ln(xn)/xn=1
This youngster has amazed me!
I'm inspired.
Congratulations.
We can also use the inverse function formula
Integral of f^-1(x)dx=
x•f^-1(x)-F(f^-1(x))+c
Could you do a proof of L'Hopital's rule? I've seen it used a lot on this channel but I don't know why it works
vdvman1
ThreeBlueOneBrown did a series on calculus. Highly recomended. In this video of that series at around 10:00 mins mark he explains why L'Hopital's rule work.
ruclips.net/video/kfF40MiS7zA/видео.html
Anshul Raman I actually have watched ThreeBlueOneBrown's calculus series, guess I missed the one on L'Hopital's rule
Or forgot about it
L'hopital wiki:
en.m.wikipedia.org/wiki/L%27Hôpital%27s_rule
It pretty much checks which grows/shrinks faster: the numerator or the denominator, by comparing their slopes (or rates of change). And the slopes are the derivatives.
that black ball is the source of all the knowledge of the universe
I really enjoyed this one. Like I said before it has been over 30 years since I was at the university but I have always enjoyed mathematics. Great job! Keep us on our toes!
Very very nice and creative solution
Damn, that kid actually left me speechless. That was amazing.
This just shows you how bad America's educational system is. Most people don't learn calculus until they're 20, and this young man is already very knowledgeable.
I'm pretty sure when i was his age, i was thinking about which dinosaur i want to be for haloween, lol. Well done.
That kid is going places. Lil genius
I've never seen such a well-drawn sign of integration
Oon Han is a smart guy. I am like twice his age and do pre-university level and I dont even learn this. Good work
Damn... this 8 year old kid knows more maths than me, Now i feel dumb haha :''D
I also wanna be a cat...
Diego Sabbagh : ))
1:40... this is like the reverse feinman technique! you just differentiate the inside and multiply by increasing n values in x^n /n!. This is golden
That kid gives me hope about the future of humanity...
I used to considere the 0.ln(0)=0, cuz anything times 0 is equal to 0, i don’t know if i was doing some sacrilege but the final answer gonna be -1 as well xD
Yeah me too
Well if you apply this to x*(1/x2) = 0*smth = 0.
But clearly x/x2 = 1/x -> inf (x->0)
You cannot say that 0 times plus or minus infinity is equal to 0 because it’s an indeterminate form, and infinity is not a number. You have to do more work when you do limits using l’Hopital’s rule.
Wow. He made the Asian cliché more true than 1=1!
This was amazing, genuinely impressed
Have people at my university who are stuck at polynomial Integrations. Seeing that a kid this young does this is just unbelievable!
Meanwhile, when I thought our math was too easy when I was young, I was told to "stop complaining and show your work" when I could solve most problems in my head.
I always loved math, but only because I was watching youtube videos with math problems slightly above my level. I really wish the 'no child left behind' policy didn't cause teachers to focus less on gifted kids. I understand the damage to people who struggle with math being left behind is greater than the damage to gifted people who don't get to be pushed ahead, but I really wish I got better guidance when I was younger.
I am so happy this kid gets to do what he loves, and I hope he gets all the further guidance he needs to keep excelling!
Oon Han all the best to you. Thank You for the informative video.
The young kid will be a good engineer
I'll be doomed.. Is that a 9-yr old kid solving integrals?
I love with your work guys ❤️
what a brilliant solution by Oon Han!
Awesome 2nd method! Fantastic! Applause!
That kid is 10 years old and solves integrals. I look at my nephew and niece which are also 10 and all they do is watch spongebob and multiply two numbers lower than 10 together. His method is explained in simple terms, making it really clever.
Hey kid he's college graduate
wow, the second method is quite simple, talented kid
The way you explained inegration by parts made my heart ache.
We can apply Feymann's technique to solve this integral.
This young man is very impressive--even if he's rehearsed. Wow.
This is an improper integral. Firstable you have to express the integral as a limit. Lim(t tends 0+) of integral from t to 1 of lnxdx
@blackpenredpen Isn't the second method like setting x = exp(u)?
His bit about making the integral negative could be improved. The reason why the integral is negative is because you’re flipping the bounds. The integral of lnx from 0 to 1 is the same as the integral of e^x from 1 to 0.
Oon Han's method is actually same as what I have thought.
Looks like that Oon Han is a kid. Do not underestimate a kid!
In his age i didnt know how to multiply damn he's good XD
Liked the 2nd method .... very clever !!!
3:33, well for negative values of x just add i*pi to ln(-x) (-x being positive since x was negative)
7:44 so the area between y=1 and ln(x) to the right of x=0 must be equal to 3, yes?
I don't like both methods. In this case I try to add an epsilon > 0 to the lower integral limit and use the limes for epsilon to zero for solving it. Integral of ln x is x(ln x - 1) and then just use the fundamental theorem of calculus and then the limes operation on it.
Question: What is the limit as n tends to infinity of (n^n)/n? Some might see it as n^(n-1) with tends to infinity but in the question you are dividing by infinity which would cause problems. @blackpenredpen
The limit would give us Inf./Inf. L'Hopital's rule would lead us to (n^n)(ln(n)+1)/1, which clearly tends to infinity as n approaches infinity.
beautifull video and the second method is awesome!!
Oon Han made it sooooo easy!
Can anyone tell me which video of his should I check out for the D-I method mentioned in the video?
I was so pleased watching this delighful video (both demonstrations), that I was about to forget to click on thumb up ;-) But it's OK, I did it !!
François Cauneau yay!!!!
Doesn't it equal lim_{n -> 0+} \int_n^1 dx/x which equals lim_{n -> 0+} (1 - 1/n) which diverges? Why does this more direct method reach a different conclusion?
Actually i was impressed by Oon after using inverses method and made my day, I subscribed to Oon. 😉 good luck
lowkey flexing supreme
Yeah, what a smart kid! How old is Oon Han? He solved this integral in a way that I' ve never thought of. Much more clever and easier.
Thank you a lot.
In another comment bprp said he was 10 ~a year ago.
I think the second way's better - certainly easier to follow! Very neat.
Amazing!!! the axis stuff!!
Good boy. Should I show this video to my 11 years old son? Will it inspire him or infuriate him? Or it will discourage him? A pedagogical question.
In Oon Han's background you can hear some really f'ing creepy sounds.
please solve: integrate 1/(xarctanx) dx from x=0 to 1
3rd method: "proof by knowing the answer"-> ln(x)dx=xln(x)-x
Suggested problem: Solve (x ^ y) = (y ^ x) for x not equal to y. I have seen this done for many X and y, (e.g. 2 and 4) but can not remember how it was solved for general X and Y. I know the graph of the solution looks like a hyperbola in the first quadrant.
Stephen Mooney Infinite solution if I remember it correctly. Have solved it way back in the past and have forgotten the answers. If you're curious in how, let y = kx where k is a real number then proceeds to solve it.
Why didn't the bounds change for the second part of the integration by parts?
I stuck in a question where I have to find integration of logx from 0 to 1 but I am in doubt because when I put the value 0 in log it goes - infinity according to graph... But you also my problems.. thank you
then why does -ve infinite comes as a result when I differentiate it further the traditional way?
I'm a big fan, and I've seen most (?) of your videos, but find myself wondering how I never saw this - Oon Han is awesome, "isn't it!"
When I was 10 I didn't know what maths are... (But now I love maths so much... anyway)
This is just about the very easiest domain to integrate for that function.
If in a definite integral we cannot integrate by parts or any other technique than can we do the mclaurin expansion of the function than integrate each new func
is there a formal proof that the second method is correct?
That editting
What a smart solution!!
I dont get why he didnt use the second method firstly, its pretty damn obvious that you could just e^x instead because that way youll be dealing with the horizontal asymptote instead of the vertical one in the logarithmic function.
In which standard that boy is?? Simply great
Increíble como un niño ya logra hacer una integral de ese nivel. Aquí en Colombia, la mayoría, solamente saben buscar vídeos infantiles en RUclips.
I have a question if the root at the bottom is the same way
I thought of that second solution but I don't have the ability to prove it formally so I decided the best course of action was to create a little slider in Desmos that rotates one function onto the other, a very satisfying and non-mathematical proof lol. Here's a link: www.desmos.com/calculator/mjyjcee8mf
good work as usual brbp but special praise to oonhan. don´t take it wrong buddy, but i liked oonhan´s method way better (it´s a simple and brilliant idea) and he explained it really well. Especially considering his age. good work young man.
I would choose substution
Oh, you're genius! I like second solution more. :)
I do too!