Complex Analysis: An Integral from @MichaelPennMath

Today, we evaluate the integral from 0 to infinity of log^2(x)/(sqrt(x)(1-x)^2).

[Decrupt]

Double the contour integrals, double the amazing content!

[wuyqrbt]

Min: 22:42
I think you need *Hagoromo Chalk*, because the *Hagoromo Chalk* are antigravity!!
Min 25:00 .... 27:00 thanks for details.
Just Great (one of the best of yours) 🤓
Thank you so much dear *QN³* ❤️

[TheRandomFool]

Very nice, this is the exact method I used when he posted that video.

[wuyqrbt]

Hello dear *QN³* ...
One integral with two methods?!
Complex one with You ( dear *QN³* ) and the other with Penn?!
*Super Duper Cool*

[kummer]

Contour integral is the real deal. I always hate when people on some books come up with the paths without explaining how they where constructed. The contour path should be selected localizing the poles in the situation. This should be explained further. The rest is easy to follow and the classical exercises of the theorem.

The deal here is how the practitioner comes up with the adequate contour and the identification of the relevant poles.

[holyshit]

Firsty I substituted y^2=x to get rid of square root then I split interval into [0, 1] and [1, infinity]
Int(8ln^2(y)/(1-y^2)^2, u=0..1)+Int(8ln^2(y)/(1-y^2)^2, u=1..infinity)
then substituted y = 1/u in the integral Int(8ln^2(y)/(1-y^2)^2, u=1..infinity)
to get 8Int((1+u^2)ln^2(u)/(1-u^2)^2, u=0..1)
Then I integrated indefinite integral by parts 8Int((1+u^2)ln^2(u)/(1-u^2)^2, u) (differentiating log)
and then calculate the limits
Finally I got -16Int(ln(u)/(1-u^2), u=0..1)
From here I expanded 1/(1-u^2) as geometric series
and calculated integral Int(u^(2n)ln(u), u=0..1) by parts
and got
16sum(1/(2n+1)^2, n=0..infinity)
16(sum(1/n^2, n=1..infinity) - sum(1/(2n)^2, n=1..infinity))
16(sum(1/n^2, n=1..infinity) - 1/4sum(1/n^2, n=1..infinity))
16*3/4sum(1/n^2, n=1..infinity)
12sum(1/n^2, n=1..infinity)
2π^2

[ameliaC-xgrg]

Great content as always :P I hope you two could make a collab video sometime, I'm sure it would be awesome

[wuyqrbt]

Hello dear *QN³*
Probably you watched the last video of Dr. Penn (looks crazy) already ....
You think its answer is right?!
Honestly, I have some problem with it. I'm not sure, but I think we have some problem on it (at least in details)!
I hope you'll have a plan for it.
Dear *QN³* ❤️

[rishetmehra]

what are branch points and branch cuts?

[soup]

hello can you do a video on the integral from 0 to infinity of cosx/(x^3+x^2+x+1) dx please!
i've tried to evaluate it by contour integration but i'm stuck

[stephyjose]

Can you help compute the integral $\int_ {- \infty} ^ {\infty} dk\frac {e^ {-a\sqrt {s+\iota k}} e^ {-a\sqrt {s-\iota k}}} {(s+\iota k) (s-\iota k)} $ using complex analysis?

[Leomenxd]

All this mess just to make two pies square shaped, damn.

[GreenMeansGOF]

Is it possible do solve integral of sin^2(x)/(1+x^2) from 1 to infinity? Wolframalpha only gives a decimal answer so I assume the answer is no.

[nasim]

How about solving

∞
∫ [ exp(-x) * ln (x) * ln (x) ] dx
0

[KStarGamer_]

There are some alternative methods on MSE for anyone interested: (I had to split up the URL because YouTube would delete my comment otherwise. Sorry about that)