Integral of (lnx)^2/(1+x^2) from zero to infinity
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- Опубликовано: 9 сен 2024
- 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...
Wow, nicely done Prime Newtons! Haha it is always rewarding and pleasure to collaborate with you on a math problem :)
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
@@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
@@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
@@holyshit922 Check Dr PK Math channel he made this channel about 10 days ago. He uses complex analysis to evaluate this integral very well
From aaalll of the math RUclipsrs ive seen, you are the only one who cares about convergence. Excelent job
This procedure is totally mind blowing.
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).
Dr PK did another complex analysis method - contour method in this video, which is pretty beautiful
I have to say - it takes a lot to get me to watch a RUclips clip longer than 10 minutes. You had me for the full half hour on this one! Bravo!!
Truthful Master, congratulations!
You are a child again...
Bless you and your beloved beings!
Thank you a lot!
Nice video prime newtons, the integral at 26.40 is a gamma function that is gamma(3).
Lot of love from Morocco
Very good video…. I am waiting for the video of Fourier transform and may be Laplace transform video…thanks
No one can beat Prime Newtons! ❤🎉😊
Haha. Thanks. I wish I was a Boxing champion
@@PrimeNewtons Maths is quite the accomplishment, sir!
Wow, great Xmas present. I love it when Black men demonstrate their genius.
There is no need to look for the integral of t^2*e^-t because this is the Gamma function. Г(3) = 2!
Yeah. Then I'd have to explain gamma function to my audience. Someday, we'd get there.
👍
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 ∞).
I wish I saw that
So I was actually looking at the derivation of the series answer and it isn't actually that complicated. You first need to transform that series into Hurwitz Zeta function form, and then use the conversion formula to transform it into a polygamma function. You can then either evaluate that immediately or use the reflection formula (which is a not well-known formula) to get the answer
Absolutely great!!!
Best!, what is the application of this equation?
you could have also used Laplace Transformation to solve the integral t^2e-t
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
Hey Prime Newtons! I got lost when you explained de substitution of x into pi/2. Am I wrong? or, is x not part of the sin function? If that is the case, then the rest of the calculation would be wrong. Please explain! Thank you
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?
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.
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. : )
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.
The n belongs inside. It eventually came back. It was a temporary misplacement. The x is inside sine
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 :)
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
What if in min 10 you push the integral to look like an
hiperbolic cosine?
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.
17:14 you are making a mistake writing n here.
непорядок, на бета и гамма функции Эйлера не ссылается? стесняется?
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.
It must be and it was intuitive :)
Could you do IMO problems?
I ran into integral which is dx/e^2x-4e^x+4. Could you do it
marvalous
Can u recommend how to move on to calculus as an absolute beginner?
Precalculus!
Khan Academy. Make sure your algebra and trig are solid first.
Can you film some geometry videos pleaseee 🙏🙏🙏
What if i put x=1/t
That works too
Put x=1/u
Your equation has letters, squiggly lines and are written all over the place. #Fake #Illuminati #Ew
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😂 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.
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P.S. RUclips 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.
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.