An Euler Experience - A dope Integral :v [ ln(x)/(x^2+1) from 0 to infinity ]

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[lukamitrovic]

"Is nothing but" *writes 10 pages of integrals*

[TheReptileDragons]

If an integral doesnt end up equalling a factor of pi or e, get it out of my face.

[PackSciences]

The integral from 1 to infinity (so minus the integral from 0 to 1) of ln(x)/(1+x^2) is called the Catalan's constant. It pops up in various "hard" integrals problems. If you want to read more about it, I recommend the article from MathWorld Wolfram about the Catalan's constant.

[EssentialsOfMath]

I used x = tan(u). Then you get integral from 0 to pi/2 of ln(tanu). Then use t = π/2 - u to get 0 to π/2 of -ln(tant), so the integral is 0.

[leonardromano]

Just substitute x=exp(y) and get an Integral of an antisymmetric function over a symmetric interval -> trivial.

[The_Professor_S_]

The Integral Bois are back
Thank you Papa
We love it RAW

[StreuB]

Wow, I absolutely loved this!!! Thank you!!! I learned several new tricks we have yet to see in Calc2!! :-D Danke!

[cadec]

I did it at my classes using complex analysis.
Using that integral(x^α/(x²+1)) from 0 to inf is equal to π/(2cos(πα/2)) [obtained with the Residue's Theorem].
We then do d[integral(x^α/(x²+1))]/dα [from 0 to inf], the differencial enters inside the integral (uniform convergence) and after deriving in both sides we get integral(x^α [from 0 to inf].
Then we make α→0 and we've got our integral :D, integral(ln(x)/(x²+1))=0 (from 0 to inf).
Hope you like this way too.

[pranayvenkatesh]

Dope. You have truly stumped me, flammy boi. Bring out the next video, Imma solve that one!

[Joshwism]

"this is just a dummy variable, you can call it your mother or this chair I'm seeing"

liked, commented, subscribed, etc.

[Riiisuu]

Wow that dog understands -nazi- German, very epic.

[amih]

Had this integral in my calculus 2 final exam 😁

[tgx]

Only one notes to integral - integral=0. We must be sure, that this integral is not infinity. For example (N) (L) integrals from x , where x is from - infinitiy till infinity don't exist

[pi]

Ah, so this one is the integral that gives the probability of me getting a girlfriend. :/

[ttnfranch]

this is so cool ! big thanks for that !!

[crosisbh]

I solved the integral of dx/(1+x^2) using partial fractions. I know it's arctan, but I wanted to try to make it fun. I factored 1+x^2 into (1-xi)(1+xi) and did the normal partial fractions and got an answer. It turns out to be the complex definition of arctan:[ 1/2i * ln|(i-x)/(i+x)| ]. You should def try it because it's neat, and over complicating a simple problem lol

[avananana]

Now this is some Quantum Information & Computation content bois.

[MrMJS-kscc]

integrate {log(1+x)}/(1+x^2)
for lower limit= e and
upper limit = e^2

[sidstam]

I feel the maths I did (probably not that well!) from over thirty years ago, start to make links again in my aging brain. Thank you.

[sebastiian]

Nothing better than spicy boy and afrojack t shirt!