This integral is actually one of your favorite constants

Complete solution development for this cool integral resulting in Apery's constant.

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Advanced MathWear:

[maths_]

If you like the videos and would like to support the channel:

You can follow me on Instagram for write ups that come in handy for my videos and DM me in case you need math help:

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[emma]

I am a lady and I watch this channel. Happy to get the mention! All three and a half of us had better have commented.

[keyaanmatin]

LOOOOL well done for avoiding the gamma function today. must have been real tough.😅

[oAstro]

Alright gentlemen and three and a half ladies,

[satyam-isical]

That switching between integral and summation operators is as beautiful as that u=(1-x)/(1+x) substitution

[CM_France]

Hi,

"ok, cool" : 0:16, 2:27,

"terribly sorry about that" : 1:36, 2:23 .

[aravindakannank.s.]

sorry mate a lil busy last two to three days so i missed out some videos i will quickly watch those tomorrow
when u tried not to let intrinsive tought by not using gamma function i can't stop laughing because of the battle ur mind and heart
finally u saved it for later video idea😂

[boltez]

This can be also done by taking
ln²1-x/1+x as t, doing some subsitutions,
then integrating by parts and applying gamma function (at least i think so).

[renerpho]

It was nice to see you go through gamma withdrawal, and make it to the other side intact.
What was even nicer is how it appears to have sparked a video idea?!

[nizogos]

By looking at the end result being (3^2)*ζ(3), I wonder if the result is always of the form n^2 *ζ(n) ( or n^(n-1) * ζ(n) ), where n is the logarithm power in the starting integral.

[mikeoffthebox]

Rearranging to ux + x + u = 1, you can see it's symmetrical, so goes the same both ways.

[wqltr]

I think the integral could also be written as a multiple of the integral of (artanhx)^3, which is pretty cool

[SnowWolfie]

Why is it not valid to simply integrate by parts where u = ln( (1-x) / (1+x) )^3 and dv = dx? After some basic simplification, when you plug in the bounds you get two ln( 0 )'s which is undefined. Also when I plug it into a not super complex calculator it also says the integral is undefined but can be approximated as the result in the video.

[johnmich]

Hi, thanks for the fantastic video, just curious what software are you using for handwriting.

[jhonnyrock]

Hello, there's this crazy integral and its answer that's just stated on Wiki without the derivation, and I was wondering if you'd be up to the challenge of solving it!

[merouan]

Thanks can you publish about PDE problems

[pluieuwu]

glad to be one of the 7/2 ladies here 😂

[slowfl]

how would you do it with gamma function tho

[MrWael]

When making geometric series -1/(1+u)^2, it should be appropriate to add 1 to the power of negative 1 in right side. So, it will be (-1)^(k). Anyhow, thank you for this interesting integral and innovative solution.

[brickie]

i feel dumb cuz idk how you would continue after invoking the gamma function. gotta watch more maths 505 videos!