Reddit Integral

[@drpeyam]

The final answer is pi^2 / 4

[@jbtechcon7434]

That's not what you said in the video. REFILM!

[@gourabghosh5574]

You confused me for 5 minutes.😡😡😡. Thnx for such an amazing video.😊😊😊

[@alexandersanchez9138]

Losing points in style!

[@JensenPlaysMC]

Why can we use this when the formula only holds true for abs x < 1?

[@98danielray]

@@JensenPlaysMC int from 0 to 1

[@AndrewDotsonvideos]

U= e^-x biggest plot twist in history

[@Rundas69420]

I had my Calc 1 exam today but also watched math content like this for like 2 years and I thought: "Boi are these integrals easy". Thats the effect of videos like that. So thanks for your content, I really enjoy it :D

[@tzovgo]

agree

[@georget8008]

Dr peyam please correct the angle of your camera with regard to the board. It is around π/6. it would be better to correct it to π/4. Thank you.

[@scarbotheblacksheep9520]

I agree, and there was some glare over some of what you wrote.

[@codingphysics695]

I like that the Taylor series of the logarithm can be simply cheated out from the geometric series. That's a real life hack!

[@nathanisbored]

I was taught that method before i was taught the more general method

[@gamedepths4792]

EZ GG 420- NO Scope !! Absolutely Demolished that integral !

[@nightish_one6007]

This might be my favorite math video ever. Like, I'm gonna re-watch this tomorrow just because of how beautiful it is! Thanks for making this video sir

[@The1RandomFool]

I attempted this before watching the video. The substitution I took was u = tanh ( x/2 ), then took a series approach to find the result in terms of the Hurwitz zeta function. This special value can be expressed in terms of the Riemann zeta function, which lead me to pi^2 / 4.

[@jayamitra4656]

Boy, you upload faster than i can watch! Great content, amazing insight!

[@helloitsme7553]

Sir you wrote π instead of π² in the end

[@drpeyam]

I did!

Yeah, he was so happy with this beauty, that he missed that little "2" at the right upper corner of pi :)

[@arnavchaturvedi4818]

Sir please make a video on Bernoulli numbers.

[@TheNachoesuncapo]

that I´d love to see

[@Absilicon]

So the sum of all reciprocal odd squares, make up 3/4 of the sum of all reciprocal squares?

[@drpeyam]

Interesting, huh?

[@Absilicon]

@@drpeyam yh it really is.

[@alichowdhury3502]

Abidur Rahman It makes sense because 1/N^2 becomes very small for large values of N and the only real contribution is 1/1^2 which is 1, and that is contributed to the odd part.

[@fenhirbr]

Thats kinda obvious, because the sum of all reciprocal even squares 1/(2n)^2 is 1/4 of the sum of all reciprocal squares

[@Absilicon]

@@fenhirbr well yh, if you do the calculation it's obvious, but if you looked at the series and compared them term by term, you can see that the odd series would be bigger- but 3 times bigger? 🤔

[@mehdisi9194]

I was surprised to see the close relationship between this two numbers namely pi and e. Each of these two numbers is dependent on a specific group of functions namely sine functions and exponential functions which are seemingly unrelated but this integral show a nice relationship between these tow numbers. Thank you so much

[@martinepstein9826]

Sine and exponential functions are not just related, they are the same thing! sin(x) = (e^(ix) - e^(-ix))/(2i) See this blog post for a more in depth explanation of the relationship between pi and e. affinemess(.)quora(.)com/What-is-math-pi-math-and-while-were-at-it-whats-math-e-math

[@mehdisi9194]

@@martinepstein9826 i know this. Euler's formula is very famous in this regard. I mean the real analysis and in my opinion, the analysis of complex numbers , although it has many beauties but it is merely a shortcut tool and I do not find a clear meaning in the complex numbers. And if you find a clear meaning in these numbers or you know more about the philosophy of introducing them or you knwo a book about the history of the emergence and evolution of these numbers please introduce it to me too.

[@whatsup7341]

Hey Dr. Peyam, please make a video on why you can play around with differentials as if they are variables instead of a part of dy/dx, especially with separable differential equations.

[@drpeyam]

Well technically you can’t...

[@whatsup7341]

Huh, so is it only in some situations that you can treat differentials as variables? In class we are just now learning about separable differential equations and my teacher just says that it's legit to multiply by dx on both sides (then integrate) in stuff like: dy/dx = 1 How do I know when treating differentials like this will yield the correct result, and when it is wrong to do. Is there some way to prove this?

[@leonardromano1491]

​@@whatsup7341 There is a theory where it gets really close to treating differentials as "variables" which is the theory of differential forms. This tool is commonly used in fields like thermodynamics, statistic physics, general relativity and differential geometry as well as multivariable calculus. However it has to be handled with care. But for example lets say that we have some differentiable function f in 2 variables f(x,y) and you want to know the differential along a way s (df/ds) then you might know df/dx and df/dy so you can write: df = (df/dx) dx + (df/dy) dy Next you know how x and y change along the way s so you know dx/ds and dx/dy so you can write: dx = (dx/ds) ds and dy = (dy/ds) ds Then from that you get: df = df/dx dx/ds ds + df/dy dy/ds ds = (df/dx dx/ds + df/dy dy/ds)ds so "dividing by ds" will give you df/ds. However this is not really what you are doing and there are some subtleties.

[@kalvin90210]

Beautiful!!! Thank you Dr. π-m

[@BornInOz]

Great video, thanks, but if I could make a suggestion, I find the camera angle makes the board very hard to read. It's especially difficult when trying to read things being written on the right hand side of the board. I find that a straight on camera angle like Pappa Flammy uses makes the entire board much more legible.

[@drpeyam]

Yeah but then with the straight board I’m covering up all my writing since I’m left-handed

[@scarbotheblacksheep9520]

But you could just move around if necessary. I couldn't read some of this at any time. So I think that would help. If they need to pause they can.

[@popalofiti480]

NOT THE BEEEEEEEEEES

[@alecvan7143]

your videos are always impressingly great! :)

[@drpeyam]

Thank you :)

[@smiley_1000]

I've also seen that one on reddit! Maybe you should link to the reddit post/comment, I don't have it anymore.

[@drpeyam]

I can’t find the link any more :/

[@smiley_1000]

@@drpeyam guess what I just found: www.overleaf.com/read/dqspcpbnqyym

[@smiley_1000]

Different way though

[@Invalid571]

Excellent video, I can't wait for the trig sub one. ☺

[@leonardromano1491]

Using this result you can get a nice identity if you solve this after substituting u=exp(x). Using some geometric series and simple integrals of the form x*exp(x) you reach at 5/12 pi^2 -(ln2)^2/2 = sum from 1 to infinity of 2^(-n)/n^2

[@drpeyam]

Beautiful!

[@Galileosays]

Cool function. The integral is finite (π²/4), while the limit of the integrand to zero and to infinity is respectively infinity and zero. The integrand has a line of symmetry at x=y. Hence, x=f(f(x)) where f(x)= ln(e^x+1/e^x-1). The shortest distance to (x,f(x))=(0,0) is at x=ln(1+sqrt(2)).

[@drpeyam]

Amazing!

[@prachisharma463]

I am not familiar eith subsituition at step2B

[@xanaxsandwich5441]

How can I stop watching these higher maths videos while dreaming about doing the same stuff and start getting ready for my high school maths which I can't do? :(

[@MathsatBondiBeach]

This type of integration problem from the 18th century was solved by Euler in his inimitable style long before 19th century concepts of uniform convergence ( so you can interchange summation and integration) or even Dirichlet’s theory on absolutely convergent infinite series. It is fascinating to go back and see how Euler actually “did the business”. Ed Sandifer has collected the Euler Archive here: eulerarchive.maa.org so interested people can go and check out how this stuff was done at the time. It is very instructive as it is the foundation of the edifice that is now taught to unsuspecting students who have no idea what the chef was actually doing to the liver in the kitchen!

[@martinepstein9826]

It would be awesome if there was a method that didn't use Taylor series as that would be a new way to solve the Basel problem! What's interesting about this integrand is that it's an involution (i.e. it's its own inverse: f(f(x)) = x). This made me think of just letting u equal the whole thing. I eventually got that the original integral is equal to the integral from 0 to infinity of x*csch(x) dx. That's a really nice bell-shaped curve so I like to think I'm getting somewhere, but for now I'm stuck.

[@johannesh7610]

I also got that after some time (x/sinh(x)), but I didn't really know how to proceed

[@GusTheWolfgang]

I tought about that as well. that would imply that the graph is symmetric about the line y = x. Perhaps we could go somewhere from there?

[@martinepstein9826]

@@GusTheWolfgang The only way I can think to use the symmetry is to divide the region into 3 pieces; a square in the lower left and two identical pieces to the right and on top. The square has a side length of ln(1+sqrt(2)) so if we could integrate from this lower bound instead of 0 then we could find the whole area easily.

[@szymon5830]

Isn't this function some kind of inverse hiperbolic cotangent in terms of e^x?

[@hOREP245]

All I know is that it is it's own inverse.

[@jkid1134]

“Oh no not the bees” really got me 😂

[@deyomash]

i tried rewrote it as ln(coth(x/2)) lol. Got stuck after another substituion!

[@JensenPlaysMC]

Can anyone explain why we can use this formula when it is defined only by abs(x) < 1 plugging x=1 would result in a divergent sum?

[@hOREP245]

I'm late to this, but for an integral to converge you only actually need (a,b) not [a,b], so you don't need to include the end points. This means we don't actually have to use 1, it is essentially acting as a supremum.

[@p12psicop]

I solved this on desmos by just using the integral calculator and guessing pi^2/4 and noticed the numbers matched closely when using limits from 0.0001 to 709. Took me 3 or 4 minutes. =P

[@mehdisi9194]

Perfect Dr.peyam👌👌👌👌 thank you so much

[@Gamma_Digamma]

Looking at it I first thought of trig substitution...

[@giuliopistolesi4969]

Splendid video !!

[@almightyhydra]

Got the A and B mixed up at 9:40. I thought also you missed a minus sign, but actually you'd done the B integral including the minus sign, so all is well. :)

[@shanmugasundaram9688]

It is hard to see what you write on the board.You see it yourself.Nice integral related to zeta function.

[@Dionisi0]

why just you didnt a w=ln(1+u) and dw=(1/u)du?

[@floydmaseda]

If w=ln(1+u), dw=1/(1+u) du, not just 1/u.

[@alexanderrey6009]

Hallo Dr. Peyam ich mag deine Videos sehr und finde deinen Enthusiasmus für Mathematik sehr ansteckend. Du motivierst mich! Allerdings finde ich die Kameraführung und der Winkel machen das Video anstrengend zu schauen. Freundliche Grüße Alexander aus der Schweiz:))

[@drpeyam]

Danke!!! Und ich weiß 😫

[@sandorszabo2470]

What is the reason of shouting? Sorry.

[@duncanw9901]

Should mention that the limits of integration are important. The taylor series only converges between 0 and 1 (probably inclusive, didn't google), so the fact that the limits of integration turned out to be that is very nice. EDIT: 1/(1+x) doesnt coverge to the taylor series at 1. It's the infamous 1,0,1.... series. Possibly problematic?

[@floydmaseda]

If it doesn't converge only at a single point, or more broadly a "set of measure zero" (cf. "Lebesgue integration"), the final result will not be affected.

[@duncanw9901]

@@floydmaseda ah. Cool

[@JensenPlaysMC]

@@floydmaseda could you explain that to me better? cant wrap my head around why

[@ojasdeshpande7296]

@@JensenPlaysMC if youre alive I'll explain

[@peppybocan]

minus one over you ;) :D

[@mariokraus6965]

Great! :-)

[@aneeshsrinivas9088]

now try using this integral to prove the basel problem.

[@drpeyam]

Ok

[@cristobalabarca2451]

I had that integral in my calc2 exam in 2015

[@aneeshsrinivas9088]

this one can be broken with complex analysis too.

[@aneeshsrinivas9088]

try substituting the whole function, to get this to be ∫_0^∞ x/sinh(x)dx

[@edificioalsacis7648]

Me encanta quensea tan feliz

[@snejpu2508]

You can also use another substitution. v = ln(1+u), then dv = 1/u du and integral of ln(1+u)/u becomes integral of v dv, which is v^2/2. Analogically for the second integral. The result = ln(1+u) + ln(1-u) = ln(1-u^2) = ln(1-e^-2x). The problem is, plugging 0 you get ln(0), which is -infinity... : (

[@floydmaseda]

If v=ln(1+u), dv=1/(1+u) du, not just 1/u.

[@rakhimondal5949]

That's the beauty of mathematics