A RIDICULOUSLY AWESOME LOG INTEGRAL: Integral of (ln(x))^2/(x^2+1) from zero to infinity

This time, we're using real analytic methods and the gamma function. Also, a very special fourier series makes a guest appearance so enjoy the video!

[zunaidparker]

Another nice one! Would like to see you tackle these problems with new techniques e.g. setting up integration in 2 and 3 variables (xy/xyz) and using polar and spherical coordinate transformations. Would be a good change and showcase even more techniques.

[daddy_myers]

Man, i was impressed by your method of evaluating the sum!


This was satisfying.

[holyshit]

Series for 1/(1+x^2) is convergent when |x|<1 so
i would substitute u=1/x in integral Int(ln^2(x)/(x^2+1), 1..infinity)
Now I would change order of summation and integration and then calculate integral by parts twice
Only one substitution, no need for function like Gamma
This sum probably also can be calculated easier (easier not equals faster)

[MovieMosaicc]

Lol Sir, Your channel is growing faster then my hairs 😂😂😂😂

[xzdvmkw]

Cette intégrale mérite d'être calculée avec le théorème des résidus

[ThAlEdison]

Have you done this with starting with I(t)=Int_0^Inf((x^t)dx/(x^2+1)), differentiating twice and evaluating at 0?

[xzdvmkw]

🗼This integral deserves to be calculated with the residue theorem

[manstuckinabox]

LOG? I'M HERE BOY DW just wanted to say thank you for introducing me to these really cool techniques to solving integrals! I'm quite disappointed that usual calc 1-2 classes usually skim past these methods...

[robertsandy]

You wrote that the absolute value of x was less than 1. However, the bounds of integration were from 0 to infinity. What have I missed here?

[ManishKumar-ichg]

Bro solve this problem if the denominator is( x^2 + x + 1)

[carlosdavid]

yo bro can you solve the integral going from 0 to infinity of (lnx)²arctan(x) ×1/(x^2+1)?
hope you enjoy this big boy!

[nicogehren]

very nice solution, but in 3:10 min how did you know the sum isnt convergent?

[asriel]

Actually the notation if ln²x is wrong, you're supposed to write (lnx)²