Integral of (lnx)^2/(1+x^2) from zero to infinity

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  • čas přidán 22. 12. 2023
  • In this video, I used mostly calculus 2 skills to evaluate this integral that typically requires complex analysis solution. @PKMath1234 is using complex analysis to solve this. Check out his amazing approach to this.
    • Power Series Polynomia...

Komentáře • 52

  • @PKMath1234
    @PKMath1234 Před 5 měsíci +37

    Wow, nicely done Prime Newtons! Haha it is always rewarding and pleasure to collaborate with you on a math problem :)

    • @holyshit922
      @holyshit922 Před 2 měsíci

      In my opinion he missed the best part
      He didnt explain how to get this Fourier series
      It is not difficult to express this integral as sum I did it in the one of the comments

    • @iqtrainer
      @iqtrainer Před 2 měsíci

      @@holyshit922 Why don't you come check the video of Dr Pk? He has a new channel, and this exact integral is posted in this channel

    • @holyshit922
      @holyshit922 Před 2 měsíci

      @@iqtrainerbecause it uses complex analysis Fourier series approach looks interesting but Newton didn't explain it
      He onle gave ready formula without proof or derivation

    • @iqtrainer
      @iqtrainer Před 2 měsíci

      @@holyshit922 Check Dr PK Math channel he made this channel about 10 days ago. He uses complex analysis to evaluate this integral very well

  • @josephmartos
    @josephmartos Před 3 měsíci +4

    From aaalll of the math CZcamsrs ive seen, you are the only one who cares about convergence. Excelent job

  • @carlosfox8201
    @carlosfox8201 Před měsícem

    Truthful Master, congratulations!
    You are a child again...
    Bless you and your beloved beings!
    Thank you a lot!

  • @stigastondogg730
    @stigastondogg730 Před 5 měsíci +6

    I have to say - it takes a lot to get me to watch a CZcams clip longer than 10 minutes. You had me for the full half hour on this one! Bravo!!

  • @rinkukoshle6537
    @rinkukoshle6537 Před 2 měsíci +1

    This procedure is totally mind blowing.

  • @maths_505
    @maths_505 Před 5 měsíci +9

    Just came across your channel. Your explanations are quite clear and excellent for students in both the academic as well as exploratory sense. I solved that bad boi a long while back using complex analysis on my channel but I like your real analytic method. An alternative would be splitting the integral into int 0 to 1 + int 1 to infinity and the transformation x to 1/x turns the 2nd integral into the first one. Then you can invoke the geometric series for 1/(1+x^2).

    • @domedebali632
      @domedebali632 Před 5 měsíci +5

      Dr PK did another complex analysis method - contour method in this video, which is pretty beautiful

  • @youssefyoussef-gp1uj
    @youssefyoussef-gp1uj Před 5 měsíci +2

    Lot of love from Morocco

  • @user-id5do9ly3z
    @user-id5do9ly3z Před 5 měsíci +2

    Very good video…. I am waiting for the video of Fourier transform and may be Laplace transform video…thanks

  • @saarike
    @saarike Před 2 měsíci

    Absolutely great!!!

  • @nikko2505
    @nikko2505 Před 5 měsíci +5

    There is no need to look for the integral of t^2*e^-t because this is the Gamma function. Г(3) = 2!

    • @PrimeNewtons
      @PrimeNewtons  Před 5 měsíci +6

      Yeah. Then I'd have to explain gamma function to my audience. Someday, we'd get there.

    • @nikko2505
      @nikko2505 Před 5 měsíci

      👍

  • @punditgi
    @punditgi Před 5 měsíci +7

    No one can beat Prime Newtons! ❤🎉😊

    • @PrimeNewtons
      @PrimeNewtons  Před 5 měsíci +3

      Haha. Thanks. I wish I was a Boxing champion

    • @punditgi
      @punditgi Před 5 měsíci +3

      @@PrimeNewtons Maths is quite the accomplishment, sir!

  • @mireyajones810
    @mireyajones810 Před 5 měsíci

    Wow, great Xmas present. I love it when Black men demonstrate their genius.

  • @ayodeleadedeji3051
    @ayodeleadedeji3051 Před 5 měsíci

    Nice video prime newtons, the integral at 26.40 is a gamma function that is gamma(3).

  • @TJStellmach
    @TJStellmach Před 5 měsíci +3

    The multiplication step at 14:25 could have been bypassed if you had folded your even function the other direction (i.e., consider the bounds of integration from -∞ to 0 instead of from 0 to ∞).

  • @g.yohannes1848
    @g.yohannes1848 Před 2 měsíci

    Best!, what is the application of this equation?

  • @nanamacapagal8342
    @nanamacapagal8342 Před 5 měsíci +4

    I have a few questions
    21:38 Why is it okay to put the n outside the sum if it's a local variable to the sum?
    6:23 Is the x supposed to be outside of the sine function or inside?

    • @chrissekely
      @chrissekely Před 5 měsíci +1

      I agree. I had the same questions. Also, given that t is a function of n, how can anything involving t be pulled outside of the summation? I'm confident that he got the right answer and his work is usually quite rigorous and not sloppy at all. However, I feel that he was a bit too sloppy this time, skipping steps and/or not explaining justifiable assumptions well enough.

    • @j4es0n
      @j4es0n Před 5 měsíci +1

      I can't speak on the first one, but for the second one the x is indeed meant to be inside the sine function. I personally do not like this notation, but it's similar to how you don't have to write 2x in parenthesis when writing sin 2x. If you wanted sine of 2 multiplied by x, you would probably just put it in front (i.e. xsin2). I think it adds a lot of confusion, especially when a question is expressed in this form. However, it does make writing down the solution much quicker. I hope this helped. : )

    • @pojuantsalo3475
      @pojuantsalo3475 Před 5 měsíci +2

      Question for 21:38 ==> It isn't, but since the term is taken inside the sum in the next step, it doesn't matter (the u² term could have been inside the sum all along before the t substitution).
      Question for 6:23 ==> Inside. The nominator should have been written sin((2n+1)x) for clarity.

    • @PrimeNewtons
      @PrimeNewtons  Před 5 měsíci +1

      The n belongs inside. It eventually came back. It was a temporary misplacement. The x is inside sine

  • @josephmartos
    @josephmartos Před 3 měsíci +1

    What if in min 10 you push the integral to look like an
    hiperbolic cosine?

  • @ThisCanNotBTheFuture
    @ThisCanNotBTheFuture Před 5 měsíci

    Early in the video you wrote "sin(2n +1)x" which confused me for a moment, because it looks like it's saying "the argument of sin is 2n + 1, then take the result of that function and multiply by x." But in fact, you mean sin((2n+1)x), correct? That's what makes the trick work.

  • @DefenderTerrarian
    @DefenderTerrarian Před 5 měsíci

    For the Fourier Sine series in the beginning, is the x inside the sine? It looks like it was outside, so I was a little confused.

  • @adwindtf
    @adwindtf Před 5 měsíci +1

    you could have also used Laplace Transformation to solve the integral t^2e-t

  • @munkhjinmunkhbayar5952
    @munkhjinmunkhbayar5952 Před 5 měsíci

    Could you do IMO problems?

  • @HashemAljifri515
    @HashemAljifri515 Před 5 měsíci

    I ran into integral which is dx/e^2x-4e^x+4. Could you do it

  • @iliyakarelin1984
    @iliyakarelin1984 Před 5 měsíci

    непорядок, на бета и гамма функции Эйлера не ссылается? стесняется?

  • @bosnbruce5837
    @bosnbruce5837 Před 5 měsíci

    1) Maple 15 returns the correct result. Although I can not make it to show step by step solution
    2) chatGPT solutions are like a conditionally convergent series...can be made to returns anything
    - 21:46 you put n outside summation :)

  • @_etg
    @_etg Před 5 měsíci

    Can u recommend how to move on to calculus as an absolute beginner?

    • @PrimeNewtons
      @PrimeNewtons  Před 5 měsíci

      Precalculus!

    • @No-cg9kj
      @No-cg9kj Před 5 měsíci

      Khan Academy. Make sure your algebra and trig are solid first.

  • @holyshit922
    @holyshit922 Před 2 měsíci

    In my opinion he should record video about how to get this Fourier sine series
    I saw video about Fourier sine series x^2 but there interval vas given and he had sin(pi*n*x) not sin((2n+1)pi*x)
    Suppose that video is watched by someone who doesn't know anything about Fourier series
    That person wouldn't knot how to get this Fourier series
    Expressing this integral as sum is not difficult

  • @sinichitaniyama
    @sinichitaniyama Před 5 měsíci

    marvalous

  • @holyshit922
    @holyshit922 Před 2 měsíci

    Prove or disprove that limit \lim_{n\to\infty} sum\limits_{k=0}^{\infty}(sin((2k+1)*x)/(2*k+1)^(2n+1)) is equal to one

  • @trollchicken
    @trollchicken Před 5 měsíci

    Can you film some geometry videos pleaseee 🙏🙏🙏

  • @kanukie_xu
    @kanukie_xu Před 5 měsíci

    What if i put x=1/t

  • @AnshTiwari11
    @AnshTiwari11 Před 5 měsíci

    Put x=1/u

  • @marcinbednara3825
    @marcinbednara3825 Před 5 měsíci +1

    17:14 you are making a mistake writing n here.

  • @vsolcarv5781
    @vsolcarv5781 Před 5 měsíci +1

    Your equation has letters, squiggly lines and are written all over the place. #Fake #Illuminati #Ew
    .
    😂 Haha I'm joking. I'm not that bright, but I can appreciate, from afar, how intelligent someone has to be to get all this.
    .
    P.S. CZcams commentators: please don't respond with something smug like "This is easy, -so and so subject- is where it gets hard". Congratulations on being big brain, but this is already beyond most people's understanding so yeah, it is actually hard and you are lucky if you find this easy.

    • @No-cg9kj
      @No-cg9kj Před 5 měsíci

      General calculus isn't hard. What dude in the video is doing is pretty advanced. You typically don't learn about Fourier series until AFTER calc 3, often in a class specifically about differential equations.
      Some people pick up on mathematics easier than others, but the vast majority of people who struggle with math simply aren't doing the work. It's a skill that takes practice and dedication. In the modern world especially, people tend to dislike things that take time and effort. People who are this good at calculus didn't just wake up one day and start integrating. He's solved thousands upon thousands of equations in the process.