So much to learn/review in one integral!!!

We evaluate an improper integral using a variety of techniques. Along the way we see:

-- u-substitution
-- integration by parts
-- transformation into a triple integral
-- geometric series
-- Riemann Zeta function
-- Dirichlet eta function
-- Dirichlet beta function
-- the Catalan constant

[cycklist]

Well I can't think of a simpler way of cleaning the blackboard.

[mythicmansam]

Came for the math, stayed for the flip.

[maxwellfeiner]

Another way to evaluate the integral of x^m log(x) from 0 to 1 is to start with the integral of x^m from 0 to 1 and then differentiate with respect to m. Fantastic video.

[billclinton]

First time I have heard of Catalan's constant. But, after looking it up on Wikipedia, fascinating how it is the solution for a wide variety of definite integrals. Great stuff!

[elyades]

Your videos are always a joy to watch. I love the subtle edits, they really flesh out your identity, and I always look forward to the next transition.
Also, I wanted to point out that you wrote "'u substituion" in the description.
Keep up the good videos !

[willyh.r.]

I do enjoy this integral problem. Please do more video M. Penn. Your work inspires most of Americans to love a real Math.

[rickshaw]

Haha, Backflip is Mr Michael's iconic action or style. In terms of backflip, you are my idol.

[victorpaesplinio]

The craziest thing about integrals is that if you know how to integrate some kind functions you can understand the resolution of some complicated integral, even if you don't have domain over all techniques.
I'm trying to pass calculus 3 this quarantine, and I have seen none about infinite series and other concepts in this video (like zeta or eta functions), but I understood very well the process. I think that in a physics course (I hope I get my physics degree soon)we don't take some topics in the "traditional order" as expected. At least I can take additional credits in math. I appreciate very much your videos!

[guilhermefranco]

Clearly he is the guy that used to do gymnastics and discovered how to use his talents on math but without getting rid of his roots. Beautiful story.

[santiospina]

Thanks for making these videos, you explain relly well and the information you give is very usefull. Keep up with the content!

[juanjosefrancocastaneda]

9:20 the most beautiful sigma I've ever seen in my whole life

[ericthegreat]

You can use the Gamma function at 7:30 by substituting u = -lnt, t = exp(-u) hence t^m = exp(-um)

[simonwillover]

Micheal Penn: To *review* about a lot of thing.
Me: To *learn* more in 10 minutes than I have learned all year at school!

[ddiq]

For the third integral, you can substitute -u=ln t and get a case of the gamma function.

[laurelcreek]

I’m exhausted just from watching Michael run this math marathon!

[kenhill]

I love these videos but sometimes little things that are dropped get to me. I screamed internally when Michael rearranged the terms of the infinite sum to simplify. The commutativity of addition is only guaranteed on finite operations. It's because the partial sums are absolutely convergent that he could lift and shift the additions of an infinite list of terms around and be guaranteed the same result. It would have been nice to have this issue addressed so that HS or UG students watching this video don't get the wrong idea that willy-nilly rearranging terms of an infinite sum is a good idea.

But, he did mention that there are theorems about rearranging the order of integration. Hopefully, this strikes the more perceptive HS and UG students to question if there are restrictions on rearranging integration are there not similar restrictions on rearranging infinite sums?

That was kind of hand-wavey reducing and doubling the terms of integration. That technique could be a video all unto itself.

Ok, toward the end of the video Michael does acknowledge the reason he can rearrange terms of an infinite summation is that the partial sums are absolutely convergent. :)

[ДенисЛогвинов-зе]

Hey, interesting math problem. Thanks. I'm interesting in knowing more about that theorem in 14:06, can you get me a link on it, please.

[jamirimaj]

ahhhh the era before "That's a good place to stop"

[henkh]

should the t at 8:27 not have to be t^(m+1) ?

[leonmozambique]

Can you explain your motivations more ? so I could realistically come up with it myself