how Richard Feynman would integrate 1/(1+x^2)^2
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- เผยแพร่เมื่อ 28 ก.ย. 2022
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We can use trig substitution (letting x=tanθ ) to do a typical calculus 2 integral, the integral of 1/(1+x^2)^2. However, we will use Feynman's technique of integration (aka Leibniz's integral rule, aka differentiation under the integral sign) to come up with a very nice integral formula.
Here's the integral of 1/(1+x^2) by using complex numbers 👉 • integral of 1/(x^2+1) ...
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He has another famous technique (at least for people who calculate Feynman diagrams) called using Feynman parameters. It’s a way of re-casting an integral you’re solving into a form with temporarily more integrals that make the original integral easier to evaluate. Of course this is only helpful if the remaining Feynman parameter integrals can be solved analytically or are at least less expensive to solve numerically (it’s usually the latter). Not sure if you’ve ever made a video on it, but in the same spirit of Feynman integration tricks!
Dude you’re still interested in symbolic calculus?
@@maalikserebryakov never know when it’ll help with an integral I’m trying to solve for research 🤷🏻♂️
yo whens the next upload
@@maalikserebryakov buddy, you're watching the wrong channel if you're not interested!
You also watch him????
Can't get enough of these integrals with Feynman's technique videos, they're just so satisfying!
P
yes!!! keep making more please!!!!
Fr ...
This kind of Integration trick is really OUT OF THE BOX, only from the brain of those genius ..... not the ordinary maths student 😆
Usually that's the case but we use these techniques after their inventions to carry on their legacy.
That's how new discoveries are made@@gagadaddy8713
That answer is really slick, because if you look at it, the first term just looks like the integral of 1/u^2, where u = 1+x^2 and then the second term is just some version of the integral of 1/(1+x^2)
Random factors of 2, I agree, but the form is pretty cool
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asnwer=1 isit
For definite integrals, I now see that there is actually no differentiating under an integral sign (requiring something like dominated convergence theorem) it's actually much prettier, we can write it as follows:
d/dx (1/a arctan(x/a))
= 1/(a^2+x^2)
Hence,
d/da d/dx (1/a arctan(x/a))
= d/da 1/(a^2+x^2).
By commutativity of partial derivatives,
d/dx d/da (1/a arctan(x/a))
= d/da 1/(a^2+x^2).
Thus, an anti derivative for
d/da (1/a arctan(x/a))
is
d/da 1/(a^2+x^2). (then work out these partial derivatives)
I recall seeing you do this in October 2018. Still a very neat video! :)
wow, what a nice way to solve this integral. Thank you for the video
Thank po sir!
I hope you will also teach this topic "definition of exp z for imaginary z" under the linear equations with constant coefficient
Didn't thought bout that amazing technique!
I learned about Feynman's trick when it came up in Howard and Sheldon's fight on tbbt, and have been stunned by it ever since.
What is that
@@SahajOp tbbt is the big bang theory, a famous american sitcom
What is the episode name?
I've tried a differents (but much longer) method
You know when you differentiate f/g you get (f'g-g'f)/g², so know it becomes a differential equation
I don't know if anyone wrote it before but if you plug in -1 for a it's going to be the same as for a=1 because of the nature of the formula, where a^3 and tan^-1 cancel the negative sign of each other.
Thank you so much, lots of love from India 🇮🇳
Rip Chen Leu. Although maybe never uttered by name again, you have a special place in all our hearts.
Bro, you're so talented, I have my undergrad as a mech E and still come to your page for fun! Please don't stop the videos lol
How’s mechanical engineering treating you 8 months later?
Symbolic calculus isn’t going to help you as an engineer you know. I bet you’re thinking of dropping out.
How are you currently doing? Just curious, good luck btw
Man that's a very toxic outlook on math and its integration (no pun intended) into engineering. I use the skills taught in diffEQ almost daily. And yeah, it was the best decision of my life to leave active duty in the military and pursue that degree. I have a solid job that keeps me entertained daily. And yeah, I'm not a quitter, hence already having the undergrad when I wrote this originally, but thanks for your concern.
@@maalikserebryakov
I'm good, love the degree. Work with a lot of EE and Physicists that treat me and my ideas with a lot of respect. @@jmz_50
@@maalikserebryakov why are you being so negative? You seem like a miserable person to be around
Excellent presentation 👌
Cool method. I did it by putting x = tan theta in 5 steps.
Thank you!
You can also solve it by substituting x=tan(u), it allows u to simplify until coming to intregral(cos^2(x)), which is easily solvable with some goniometric formulas.
Correct
Did the Same thing :)
Man, that was beautiful!
Great 👌 video , much helpful!
This video made my day🔥
absolutely fantastic
i love you man, you are very carismatic even not trying it
Great video as usual! On 0:49 You forgot to square the constant c ;) ;) ;) ;)
If Bro can make an entire playlist on feynman's technique : I one over zero percent sure I will watch it completly.
Love this!
That’s wild!
This video has an innovative new method of solving such integrals. Here is the old boring way for the same -
set x = tanT which changes the problem to INT { cos^2 T = (1+cos2T)/2 } dT = T/2 + sin2T/4 = T/2 + 2tanT / 4sec^2 T = [ arctan x + x / (1+x^2) ] / 2
We indians flooded everywhere 🤣🤣
@@AyushGupta-cj3sy This Equation says about feyman technique
@@subramanyakarthik5843 buddy i mean to says indian 🇮🇳could easily solve these
@@subramanyakarthik5843 I understand
Bhai
But it's mostly in their higher studies
But we have in 12
@@AyushGupta-cj3sy Hey im an IT professional i can solve these problems easily just exploring at these logics here
I have no idea what any of this is, but it’s fun to watch
Thanks
What a nice idea to integreat the seemingly impossible func
Bruh this was literally a question in my homework today
Thank you sir
"Why do we add the +C at the end?" It depends on what you consider integration to be. Normally we just think of integration as the opposite of differentiation. But then, what is differentiation?
If you think of differentiation as a function from functions to functions, then integration should be its inverse function. But there isn't in general a left inverse for differentiation, because it's not one-to-one - and there are multiple right inverses. So you might consider "integration" to be the entire set/class of right inverses of differentiation - such that whenever you compose "integration"/differentiation, you pull back this abstract layer of set/class and compose them with every instance of an integration function. So differentiation after "integration" is just the set/class of differentiation after right inverses of differentiation - which all collapse to the identity. And there's the added bonus that with just a little more information (such as a single point on the curve) you'll be able to choose one of those integration functions to "act" as a left inverse for a specific input - so the whole set/class of integration functions can act as a left/right inverse for differentiation.
For single variable calculus, that's about all you need to consider, and this is a perfectly fine way to define the integration notation. For multivariable calculus, there's a new wrinkle. You can have a function that's constant in one variable, but not another (Let f(x,y) = y, then d/dx (f(x,y)) = 0). So if you integrate a function in the variable x, then you pick up a constant in the variable x. And then if you differentiate that by the variable a, it doesn't always become 0, because "constant in the variable x" doesn't imply "constant in the variable a". Sure, some functions are constant in both x and a, but not all. So if we compose differentiation with "integration", some of those compositions will collapse the constant, but not all. We didn't add +C at the end, it never should have been removed.
This is another approach: th-cam.com/video/dbNhrrWUkkc/w-d-xo.html
Brilliant
First time seeing this trick
Nice! The caption of the video could have been "Integrating a function without integration"
One mistake: C is a constant in terms of x not in a. Hence, the partial derivative d/da C is not zero in general, it is just another constant in terms of x (which you added back in the end). Nice video! (PLEASE SEE EDITS BELLOW BEFORE YOU COMMENT)
Edit: C may depend on parameter a however it wants. Thereby, C can be any function of parameter/variable a, and so, it might not be differentiable with respect to a. So in the end, the best way is just to find one antiderivative of ∫1/(a^2+x^2)^2dx. Then when we set a=1, we now know one antiderivative of ∫1/(1+x^2)^2dx. But we know that all the other antiderivatives of ∫1/(1+x^2)^2dx are obtained by adding a real constant to the antiderivative we already know, since g'(x)=0 for all real x if and only if g is a constant function. This is basically what @blackpenredpen did, but the reasoning that C is a constant with respect to a is not right although it is irrelevant mistake for the main point of the video.
Edit 2: First of all my claim is that ∫1/(a^2+x^2)dx=(1/a)arctan(x/a)+C(a) where C:R->R is any function and R is the set of real numbers. If you think that I'm wrong and that ∫1/(a^2+x^2)dx=(1/a)arctan(x/a)+c, only when c is just any real number, i.e. constant with respect to both a and x. Then your claim against my claim is that if C:R->R is not a constant function, then
(1/a)arctan(x/a)+C(a) is not an antiderivative of 1/(a^2+x^2) with respect to x. To help you, I will go through what you are trying to prove and why it is not true.
So you need to take a function C:R->R which is not constant, for example you can think C(a)=a. Then you need to prove that (1/a)arctan(x/a)+C(a) is not an antiderivative of 1/(a^2+x^2) with respect to x. But you know the definition of antiderivative for multi variable functions, so you know that by the definition you need to prove that the partial derivative d/dx ( (1/a)arctan(x/a)+C(a) ) is not equal to 1/(a^2+x^2). But we know the following partial derivatives: d/dx (1/a)arctan(x/a)=1/(a^2+x^2) and d/dx C(a) = 0. So by the linearity of partial derivative you have d/dx ( (1/a)arctan(x/a)+C(a) )=1/(a^2+x^2). Thus, your claim is wrong and we have ended up proving that (1/a)arctan(x/a)+C(a) is an antiderivative of 1/(a^2+x^2) with respect to x if C:R->R is any function. In the comments you can find also different reasonings and how other people realized this.
If you still disagree, please read the 50+ other comments in detail, read my arguments, read others arguments, read why in the end they realized that C can be a function of a. Our comments are not the best source so I also recommend studying or recalling multivariable calculus and before that one variable calculus. Even better is to go to talk people in some university's math department. If after this you still feel that I'm wrong, then G I M M E A V A L I D P R O O F of the direction you are claiming and cite to my previous comments and show where I went wrong so the conversation is easier and faster.
A constant is a constant, independent of any variable... That's what I think. So it's pretty much staright, no mistake
@@tobechukwublessed4274 Yes you are correct, but here C is only a constant in terms of x. So first we were just in the "x-world" where C is just a constant. But when we introduce the parameter aka new real variable a we are not anymore in the "x-world", we are in the "xa-world" where C could depend on a while it does not depend on x. It might sound nit picking but this is really important in multivariable calculus.
One real variable: Here when we talk about integration in one variable we mean antiderivative aka inverse derivative aka indefinite integral, i.e. that if F'(x)=f(x), then ∫ f(x)dx=F(x)+C, where C is just a constant aka a real number. So the antiderivative ∫ f(x)dx gives the set of all functions F whose derivative is f.
Two real variables: In this video to do the Feynman's trick our function depends on two variables, namely x and a. We want to have the same property as in one variable, that is, ∫f(x,a)dx gives the set of all functions F whose partial derivative with respect to x is f. Thereby, if d/dx F(x,a)=f(x,a), then ∫f(x,a)dx=F(x,a)+C(a), where C is now a real function, which could be a constant. Suppose that we allow C to be only a constant, this is a valid definition but not very useful which I try to clarify by the next example. Let F(x,a)=x+a. Then d/dx F(x,a)=1, and so, ∫d/dx F(x,a)dx = ∫1dx=x+C. If we don't allow C to be a function of a we don't have the nice antiderivative property mentioned above, i.e. F(x,a) do not belong to the set of functions obtained from the antiderivative ∫d/dx F(x,a)dx. More practical example why we want that the antiderivative property is satisfied is that we want to have working tool to solve partial differential equations.
Also @Phoenix Fire has really nice comment also in this comment section which clarifies this thing.
Hopefully this clarifies.
@@SakariPirnes it's a cool observation, it really reminds me of solving ODE's by method of Exact equations where some constant may pertain to some variable after integration. But... If you examine carefully what he did, he differntiated both sides partially with respect to a... Now in partial differentiation, the only thing that is permitted to stand is the variable which we are differentiating with respect to, all other variables and/or constants will be assumed as constants for the time being, and what happens when we differntiate constants?... They vanish!
But When we integrate partially, a constant function of the other variables apart from the one we integrate with respect to comes in place.... For instance.... integrate an f'(x,y,z) partially with respect to x will yield f(x,y,z) + h(y) + g(z) where h(y) and g(z) is any function of y and z respectively, be it a constant function or any other type.... But when we differntiate partially, all constants what so ever must vanish. That's my point...
Finally, he was integrating, yes, but he differentiated while integrating, that's the beauty of that method. It's stainless
Metodo interessante,bravo!!!
wow my math guy🔥👊💪
i thought U gonna do IBP w/ DI method😁.
By the way 4:57 the constant C, it does not depend on x, but might be depend on a (Like in an Exact ODE solving procedure).
Thus technically, when differentiate WRT a, we should have C'(a) which is another constant that des not depend on x,
and eventually U will rename the last integral constant as C or c or whatvever u wanna
😁
Yeah, I had to pause at 7:04 when he added the C back. Nope, C should not have been removed. It is constant w.r.t. x not a.
@@BetaKejaC is technically the arbitrary constant of integration which is just a variable number.
But it is a number after all and thus differentiation of any number with respect to anything is 0
No hay duda de que Feynman era un genio. Gran video, saludos desde Santa Marta, Colombia
Wow. Why did I never think of this?!?
man of action.
Excellent.
一个小小的建议，您可以在视频的演算完成后留一两秒左右方便截图，视频很棒，感谢您的付出！
好, 謝謝!
btw, 這部也有中文版的 (自己幫自己打廣告 哈哈) th-cam.com/video/sh5dQ1ZxfTY/w-d-xo.html
@@blackpenredpen You need to take a read at the above comments made by @Sakari Pirnes . Some of those here are professional grad students who have gone through real analysis and functional analysis. In your video you made a mistake by considering C as a normal constant treated the same as that in single variable calculus. But the C here is not an ordinary C; it is a function of 'a' not necessarily zero after being differentiated with respect to the parameter 'a'.
Qué genio eres, y Feynman también.
you can use chutur putur points here
like take x common from the bracket, it will come out xsquare and inside the bracket call it t and then boom
only lakshiyans will understan
thanks bro💖🌺
Very cool
👍 thank you
What is the rule of differentiating the integrand inside the integral?
That's so cool
It's so reminiscent of eigenvalue equations
Just substitute x=tan theta you will get theta/2 + sin2theta / 4 where tantheta =x
Good
Best !!! Thanks
Calculus, the most powerful mathematics in the world and it would blow your mind clean off, you've gotta ask yourself one question: "Can I integrate? Well, can ya, punk?"
物理里这种操作真的多，代数求和或者积分结构加偏导，真的是很漂亮的做法
然而他在影片裏展示了錯誤的訊息，他將C看成了一個普通的常數，但C并不是一個普通的常數而是一個C(a)的實變函數，他在影片裏把它看作了C(x)因此直接偏導得零，這是一個數學裏很大的錯誤（即使他很幸運地得到了正確的結果），他思考的不夠嚴謹，沒有往多變數微積分的方向去思考。如果他瞭解汎函分析和實變函數論，他就不會犯這個錯誤。倘若你用已知 ∫ 1/√(a² + x²) dx = ln ∣x+√(a² + x²)∣ + C 的訊息去用他上述的方式解 ∫ 1/∛(a² + x²) dx 你就會知道我的意思了，不會得到正確的答案。費曼的方法其實就是萊布尼兹的積分法則，但萊布尼兹法是建立在有界的積分上，并且要遵守收斂定理，不可隨意亂用因爲那是有局限性的。這裏的作者拿它來解不定積分就是一個錯誤的做法了。這個如果不去做深入的講解會誤人子弟的。數學是一個非常嚴謹的學科，不能有漏洞的。
@@pashaw8380 多谢老哥指点，我并非数学和物理出身，做的也是一些无关紧要的脏活，很多东西确实不太了解。
@@pashaw8380 对于泛函，我就在分析力学，qm以及工程数学中有浅薄的了解，推导和证明是很快乐的事情，但更多拿来算。
This seems easier then trig substitution
Sir more good problems 👍👍
You could just solve it by substitution taking x= tanθ and dx = sec^2θ now in the denominator (1+tan^2θ) = sec^4θ now in the end we get cos^2θ and using cos 2θ formula we get θ/2 +cos2θ/4 now by subst. in the end we get tan-1x/2 + 0.5((x^2)/(1+x^2))
simplest method i could have thought about .Takes about 2 minutes to solve .
I think it pretty good... and as you end with, cool!
I came here after Howard mentioned about it in TBBT to an answer to Sheldon's question.
Magician of parameters
Amazing
Bro!! you could have done this by just substituting "x" as "tan theta", which will give you integration of Cosine square theta w.r.t theta and that's very easy to solve without using Feynman's Technique!! Btw the Feynman's technique is excellent!!😁
Is it legitimate to use Feyman's trick with indefinite integrals? An indefinite integral ∫ 1/(a²+x²) dx is in fact 1/a arctan(x/a) + C(a), where C(a) is not just a constant, but any function depending on 'a', but not on 'x'. Then the derivative d/da has an additional term C'(a) which is impossible to find ! In this case it's a mere coincidence that result is correct (if it really is).
Take another example F(a)= ∫(x^a) dx =x^(a+1)/(a+1)+C(a). If you ignore C'(a), differentiate F'(a)=∫(x^a) ln(x) dx = x^a - x^(a+1)/(a+1)², then let a=0, you get ∫lnx dx = 1-x, while the correct answer is ∫ lnx dx = xlnx - x + C.
BTW, I don't believe Feyman himself ever used his trick with indefinite integrals, so the title of the video looks misleading to me. 😊
Feynman is here! ... Cool
I love Feynman (he's my favorite scientist ever).
Feynman was a crazyyyy dude
Can you use same technique to integrate 1/(x^2+x+1)^2 ?
Great 👍
Hi BPRP!
Hiii
I used regular trig sub and got the same answer...
Me: (10 years since doing calc during first 10 seconds of the video): *slaps head* "of course!"
Please solve one integral .... integral from 0 to π dx/ (square of a square cosine square x+ bsquare sine square x)
Yo can you make a video on how to get better at math? Ive been struggling for so long
Feynman was the man!
sir please of this:
integral of sqrt(x^3+1) w.r.t dx
Just cool
So does it wind up not mattering that the value of c, for any constant boundary conditions, is variable with respect to changing the value of a???
Integration of 1/(1+x⁴) please
I got the answer as 1/4 * Sin(2*Tan-1(x)) + 1/2 * (Tan-1(x)) by logical way
You can just do this with partial fractions.
can you make a video of limit integration?
What if we substitute x = tan a ,
Then , dx = sec² a da
Substituting in above
1/(1+x²)² dx
→ sec²a/(1+tan²a)² da
→ sec²a/(sec²a)² da
→ 1/sec²a da
→ cos²a da
→ 1/2 × 2cos²a da
→ 1/2 × (1+cos 2a) da
Integrating we get ,
1/2 (a + (1/2) sin 2a) + c
Now, since x = tan a
And we know sin 2a = 2tan a/(1+tan²a)
Therefore, sin 2a = 2x/(1+x²)
→ 1/2 ( tan^-1 x + (x/1+x²)) + c
That's the answer....
I love it
wow great video! and fun way to solve it
Thank you 🤗
yo please if i want to ask a question can i ask you and how to ask
thanks
Itu jenis intgral pecah rasionak. Maka dimisalkan
1/(1+x^2)= {(Ax+B)/(1+x^2)}+{(Cx+D)/(1+x^2)^2}
Dari asumsi tsb, konstanta2 ABCA dapat ditemukan
1=(Ax+B)(1+x^2)+(Cx+D)
1=Ax+B+Ax^3+Bx^2+Cx+D
1=(B+D)+(A+C)x+Bx^2+Ax^3
Yg berarti A=B=0; D=1; C=0
Int menjadi
damn why dont they teach these bangers at school, this is so coool
Məşədi qardaslarim hazir olun artiq döyüşə
Please more often new videos)
How do you integrate 1/(x^a+a^x)?
Can we do this by x=tantheta and then diffrentiatte that functin w.r.t to x then putting the value of dx through this and then final it become (cos)²theta dtheta ??? Can we do this
i'm a simple man. i see feynman i click
Probably this technique was known long before Feynman. I used to use it not knowing that
it was his technique.
just solved this question in my JEE online course.....
i also
Feynman technique of integration is OP!!
known for more than two centuries before Feynman
Why can't we use partial fraction method ??
Is that a Michael Penn T shirt?