An integral starring Gauss!

This integral has a bunch of x and ln(x) in it...but Gauss is hiding in there after a little substitution!

New math videos every Monday and Friday. Subscribe to make sure you see them!

[vivekjha]

Great video. Unsure if you've taken a math-stats course yet but another "shortcut" method is this:

After you complete the square @ 3:44 you can let U have a Normal Distribution with mean = 1 and variance = 1 / 2.
Then the pdf of U is g(u) = exp(-(u - mean)^2 / (2 * variance)) / sqrt(2 * pi * variance).
This simplifies to g(u) = exp(-(u - 1)^2) / sqrt(pi).
So your original integral becomes: I = e * sqrt(pi) * integral(u * g(u), -inf, +inf).
But one may recognize that integral(u * g(u), -inf, +inf) is the definition of the mean of U which we constructed to be 1.
Thus our original integral becomes I = e * sqrt(pi).

^ Honestly uses the same techniques in it's derivation, but maybe faster to remember this if encountered in a competition (this is debatable though lol)

[malharsingh]

Ah how often is it that an integral equals 3^3/2

[wuyqrbt]

Even for second time, this video is super cool, with its beautiful result.
Again, Thank you.

[noway]

I love it when pi comes out of nowhere.

[vcubingx]

Was not expecting that s/o but thanks! The video was awesome!

[bmt_techno]

Cool video. I liked how concise and easy to understand that was

[abdeljalilpr]

Please do another video about linear algebra and matrix in a 3d space

[tiziocaio]

Right after I asked you about this integral, I was able to figure out the answer. I thought it was way more difficult, but in fact it was pretty easy. Btw I always appreciate your explanations.

[AlgeArid]

I tried substituting in w=u^2-2u, and for some reason it results in sqrt(pi)/2. I have no idea what I've done wrong.

[HeyKevinYT]

I love when famous constants pop out of nowhere, like pi 😂

[tgx]

I had a similar idea, only longer. I  used  method by parts for integral (x^2)*([exp((-ln^2) x) *( -2 ln x * 1/x]), then the substitution y= ln x as you, and Gauss

[EpicFishStudio]

*_(teacher 15 mins of algebra, 3 different substitutions, 1 random proof and 2 group theory references later)_*
teacher: yeah so what we end up in is a special function integral that we are not going to get into here on but i let you know the answer is actually η((-7/11)^2-1) where mu is euler mascheroni constant! yeah yes isnt that satisfying answer??? i think it really is!! o-uhm is there really no questions...?

[ahmedmukhtar]

What does his shirt say? Never been a fan of these notations

[sergiokorochinsky]

How about x^p instead of just x? More interesting and general result...

[nailabenali]

Heey I just found your channel and it's very interesting! Do you have something about a topic called "summable families"? I don't know if it's the right name in english (I'm a french student!) Keep up the great work!

[isaacstamper]

What level of schooling are you at? Late undergrad, I’m guessing?

[Archipelago.]

I just want to meet this guy 😄 !

But I'm here in india on the other part of ocean 🙁