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That's funny. I solved this problem while in my second year of undergrad. I was at my friend's place and we had ordered pizza. He had just finished taking his first course on Calculus and I joked that he couldn't find the derivative of x^x, which he did. Then he challenged me to find the integral, which I did too. It was fun!

Where’s the Family Guy and the Soap Cutting videos on the sides? Please consider adding them next video!

1-1/4+1/27-1/256+1/3125-1/6^6...

👏👏👏🔝

@ 10:02 I think it should be Gamma(n+1) not Gamma(n-1)

3..

Could you prove that the last part converges ?

5:00 let y=x lnx, you say that lim[n to inf] (y^n/n!) = y. I don't believe that, I think it's 0. Wolfram alpha too.

I am happy there is no music (but no need to be so angry imho). Wonderfully explained but just a gentle constructive comment is maybe try to write a little more clearly. I know what is going on if I listen along but if I am just looking at the integral I start thinking what is planck's constant popping up in inappropriate place. Haha - just kidding - but you wrote the k really well once so I know you can do it :)

I could solve this integral in the bat of an eye. I chose not to 😎

I'm not exactly opposed to differentiation under the integral sign, but there are conditions that have to be checked if we're doing mathematics (rather than formal manipulations that may or may not make sense). I find it easier to keep track of the conditions for Fubini's theorem (although the conditions for switching integration & differentiation may be equivalent). Here we have a positive integrand (a e^{-a^2(1+x^2)}), so integrating out x & then a or vice versa will give the same answer. 'Destroy' is an over-statement. The approach seems to me no more elegant than the usual polar-coordinates trick. I also suspect that knowing the answer already helped Feynman to find this path to calculate it. On the other hand, knowing the answer and having seen Feynman's solution to two other integrals (although I had also seen one of those presented using Fubini's theorem), I still had no clue what the trick would be in this case. I tried some ways of writing e^{-x^2} as an integral, but nothing helped.

Integral is a continuous sum , we solve it by finding the value of that sum but you transfer it to a discreet sum

The final result is mind blowing great work.😊

Suggestion for future videos: Do not write the plus sign like a "t", specially if you are going to use the variable t.

Hello dude, Nice vidéo.. what app did u use in this video?

You are a gentleman and a scholar, sir. Apology accepted. I personally don't want music to distract my thought process when doing math. Absolutely amazing solution.

0:37 I'll give you a second[INSTANT JUMP CUT]Right!

Hey! I was a bit confused on the very last part. Could you please explain how you rounded the 'sum of alternating inverse squares' to roughly 0.783431. From my limited understanding, the sum should approach (pi^2)/12, which is approximately 0.822467. Thanks for the awesome video either way!!

I think this is wrong. n! = Gamma (n+1), not Gamma (n-1).

20:35 not sure if that step is valid. the numerator inside the integral is NOT exp[ 0 x constant ], it is exp [ zero x infinity ] gauss is rolling in his grave, to see mathematical rules treated so cavalierly.

suggestion: the only people who will be interested in this are likely to have at least one year of university calculus. however, your presentation is done at the level of quite introductory calculus. that's a mismatch. cut out some of the really basic stuff.

Isn’t limit n->infinity of fn(x)=0 for the first condition of the Dominating Convergence theorem? It doesn’t converge to xlnx

Your "u" looks an awful lot like your "k", leading to confusion.

Really interesting result but a laboured path through a lot of obvious steps.

Good work dude!! This is one of my favorite integrals and i really liked your explanation on how to solve it. Hoping for more to come! BTW, your voice is very soothing

Bro we cannot expand it about x = 0.

the gamma function is incredibly cool! love the videos!

good math fun...

thank you so much!!

amazing thank you so much!! 😊

Absolutely brillant ! Thank you a lot

Well done. Loved the music. It is slow because explanation are supposed to be slow, you can always fast forward it, but a fast presentation will be difficult to understand for many, with no easy fix.

I loved the video and everything, but there's one thing that I would like clarification about. Isn't there any condition on the function that needs to be met in order to interchange the integral and the summation? And if so, was it satisfied here? Thank yih

Work of art

What answer did official Putnam authority give. Was it this one?

Very nice presentation. Thanks.

Perhaps rewrite this integral as being the integration from 0 to 1: x^(i^2 * x) dx Maybe this might help. We can think of the exponent: i^2 * x as being periodic where x is the multiple or enumerated counts of a rotation by 180 degrees or PI radians. If we look at i^2 as being sqrt(-1) * sqrt(-1) = -1. We know that the value of -1 is a 180-degree rotation of 1. We can use this along with the trigonometric properties to help solve this. From what I can see, there is no one exact result from this definite integral. In other words, it's not going to give you a discrete area, volume or region under a curve or manifold. It appears that it is going to give you a series or sequence of them with a specific periodicity determined by the magnitude of x within the exponential component.

Thank you for the excellent demonstration and...NO MUSIC is perfect so one can really concentrate🙂😂

When I actually went to school, I was quite bothered by things that were distracting or incomplete, for a grade was going to come to hold me responsible for it just the same. Now, nobody is grading me on following it. Still, some others may be trying to keep up with school, and bringing their concerns with them, and perhaps a MITE irritated with the whole thing, because a grade is going to result at the end. I will not cuss you out, but I will urge you to maximize the helpfulness. k is k. It is not h, it is not u. And if on top of it, some random music is coming, this makes it harder. (Turning the sound off is not a good option, for you are speaking.) Thank you very much for understanding. It's not all about you either.

When learning any practical skills by repeating the rules harmonically to position cause-effect functions in numberness dominance sequences of positioning, (One Electron Theory Wave-packaging guesstimate), it is absolutely always NOW, e-Pi-i sync-duration instantaneously and the Universal state-ment of 1-0-infinity conformity to the Singularity-point Logarithmic Centre of Time Duration Timing, aka QM-TIME Completeness cause-effect Actuality. The exercise demonstrates how i-reflection rotation reduces 0-1-2-ness in 3-ness to the method of reverse-inverted Pi-bifurcation function. Physics combines nodal-vibrational point-line-circle strings and drum head mass-energy-momentum continuous unity-connection categorizations. From self-defining experience of embodiment, we extract the information of In-form-ation substantiation holography vanishing-into-no-thing Perspective Principle. Good practices.

I'm not a mathematician. Is it appropriate to use height of human adults as an example of a normal distribution which is symmetric about the peak, and extends from negative to positive infinity. Height is restricted to positive numbers, and is not symmetric about the peak. Furthermore, humans have two sub-populations: men and women who have different average heights. Is it appropriate to lump them together? Also, for many of us, it's impossible to learn math and listen to music at the same time, please pick one or the other.

Have a small question, isnt -a² (1+x²) = -a² -a²x² ? He wrote -a²x² Pls let me know if im right or wrong

Is this actually a Putnam question? What's the problem statement? It's a very cool result but how would a Putnam test taker know that the answer is simplified enough at the end, since it can't be evaluated exactly?

Woa very cool result. Awesome vid

I would say the problem solved in the video looks harder than the question asked in the actual exam. Namely, asking: "find this integral " does not give you any clues. On the other hand, asking: "prove that this integral equals to this sum" is quite a clue. So the video is a bit misleading/clickbaiting.

Gave it a try, but the music is very distracting. Using closed captioning helped, but not really worth the effort when I wanted to actually learn something Feynman today

Liked the music but perhaps a little quieter. I really enjoyed this presentation. Thanks.

Actually you didn't explain why you could interchange the integral and the sum. This is due to the dominated convergence theorem. We must explain all the details why we can do it