A cool integral for Apery's constant (ζ(3)): int 0 to 1 (x(1-x))/sin(πx)

[@The1RandomFool]

Before watching the video, the substitution I made was pi/2*x = u, then used the double angle formula for sin in the denominator of the integrand. In the numerator, I used the fact that sin^2 u + cos^2 u = 1, then integrated by parts.

[@monikaherath7505]

Could you write it out please I don't understand how you got to sin^2 u + cos*2 u

[@The1RandomFool]

sin^2 u + cos^2 u = 1 is a basic trigonometric identity. In general, it's a strategy to cancel out a product of sines and cosines in the denominator by setting 1 = sin^2 u + cos^2 u in the numerator.@@monikaherath7505

[@CamiKite]

Using Euler reflexion formula you can also show that 7ζ(3)/pi^2 = integral(gamma(3/2-x)*gamma(3/2+x)) from -1/2 to 1/2. It doesn't help to integrate, but it's beautiful ;-)

[@manstuckinabox3679]

Bro was like LOL do it by parts and stuff but really didn't bother by going the distance, great video as always brother.

[@maths_505]

Thanks mate. I've been looking for an integral to use that series on

[@carlosgiovanardi8197]

GREAT!! An interesting class of integrals: integrate from 0 to 1 [ x^n (1-x)^m / sin(πx)].

[@gjjkhjkk9241]

Hi, use the properties of the gamma function like is relation to 1/sin(x.pi) and x(x-1)

[@trelosyiaellinika]

Exhilaratingly beautiful!

[@MrWael1970]

Thank you for this innovative integration. Smart solution.

[@KalininEvgen]

It looks like it can be generalised. If n=3, then ans can be written as (2n+1)zeta(n)/pi^{n}. May be exist some types of integrals which are generalised that way?

[@austin4768]

Yes I’m also curious if there are analogous formulas for zeta of odd (or even all) n. One can go try to go back through the calculation and find a point where the integrand can be tweaked in such a way to yield this kind of formula. I’m too lazy to do this right now though

[@quentinrenon9876]

Nice one ! I now challenge you to solve the integral from 0 to pi of x * (sinx)^n dx. It took me around an hour or 2 to solve and is absoluetly gorgeous

[@grigoriefimovitchrasputin5442]

Thanks ! It looks interesting to solve

[@quentinrenon9876]

@@grigoriefimovitchrasputin5442 Tbh it's not too difficult but you need to be creative, normal methods don't work. Or at least I couldn't get very far using them

[@grigoriefimovitchrasputin5442]

@@quentinrenon9876 it looks like Wallis integrals, except that there is a x. Anyway, i'll give it a try

[@alarka1782]

I think by applying property of integral, it breaks into pi * integral 0 to pi/2 (sin x) ^ n dx. Should be easy from there to express in terms of gamma function.

[@ガアラ-h3h]

Should be doable by integrating by x = sqrt(x) then u = x/2 and then letting u be sin x which leads to a structure very similar to the beta function can’t rlly go further because I have no pen rn

[@bigbrewer3375]

damn, maths in 4k is fancy

[@julianwang7987]

Isn't the essence of the proof the following Fourier Expansion for csc(x) = 2 SUM_odd (sin(nx))?

[@gregsarnecki7581]

Awesome integral!

[@bartekabuz855]

Bring back dark thumbnail 1/4

[@maths_505]

Okay tomorrow

[@PhilesArt]

cr7 fan here too xD, loved that one 👍

[@maths_505]

SUIIIIIIIIIIIIIIIIIIIII

[@damrgee8279]

How is this applied to every day life?

[@insouciantFox]

Integrals are everyday life

[@daddy_myers]

Integrals are life.

[@damrgee8279]

@@daddy_myers example please

[@daddy_myers]

@@damrgee8279 Your question alone indicates you could use more integrals in your daily life.

[@damrgee8279]

@Jacques-kc4qy it’s amazing that people like yourself are incredibly smart when it comes to this stuff I don’t understand any of it, But yet when it comes to political leanings regarding logic and common sense we have some of the stupidest people on the planet

[@natepolidoro4565]

Didn't go the way I thought

[@giuseppemalaguti435]

Corretto, ❤ho usato sinx, formula esponenziale, poi la serie geometrica... Anche se a me risulta un segno -... Ah ah, al primo colpo non mi viene mai... Ok, il segno - è scomparso... Is correct

[@physicsiseasy2623]

I love your solution though I don't understand one of your solution. I'm an A level student.😊

[@maths_505]

Oh cool....I used to tutor A level students..... Thanks bro....you'll understand them all pretty soon cuz you'll study the basics in your A levels....then you'll solve these integrals and DEs on your own.