This wacky integral has a beautiful result

I think this is an integral that Euler himself would be proud of. Solving it is a pretty wild ride with lots of cool techniques invoked.
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[@justcuzy3673]

*pi(e-1), as k sum started at 1 instead of 0.

[@Spiderp-p1l]

*thankfully* you're mistaken, he just accidentally wrote a 1 instead of 0(trace back to the double sum)

[@hizon525]

Yeah, happens at 9:49

[@sadboy4today187]

I love watching these videos as they satisfy my math brain needs. Keep up the good work man 👍

[@xizar0rg]

So much of "we're on the right track if it looks cool" comes from being able to recognize the patterns, which comes from practice with the patterns. (Also, I'm very sure the author is aware of many integrals where "it looks cool" takes you down the wrong rabbit holes.)

[@maths_505]

I'm making a video related to that concept tomorrow 😭

[@ericthegreat7805]

Math is just magic

[@xanterrx9741]

Beautiful Result thanks for making video

[@CM63_France]

Hi,
"ok, cool" : 2:16 , 6:06 , 7:30 , 8:05 ,
"terribly sorry about that" : 3:06 , 3:51 , 8:23 , 10:21 , 11:16 .

[@GeoPeron]

Ah yes, pie, served warm and tasty like some oily maccaroni.

[@Anonymous-Indian..2003]

Me used f(z) = eᶻ / (z - 1)
Contour |z| = 1
And then find half residue.

[@julianwang7987]

At 8:10, sin((2k+1)x)/sin(x) should have equaled 1 + 2 * sum_(n=1)^k{cos(2nx)}

[@achrafsaadali7459]

I was about to comment this mistake as well

[@thegoofiestgoooberr]

fortunately, the answer was still correct as the term ended up going to zero anyway once the bounds of integration were applied

[@MrWael1970]

Very nice solution, but the result should equal to ((pi*e/2)-1). This is due to starting index variable k from 1. Thank you.

[@shanmugasundaram9688]

There is an error in writing the Dirichelt kernel.But it never changes the result.

[@slavinojunepri7648]

Fantastic

[@chaosredefined3834]

So, at the end, we still don't know if it's rational or irrational :P

[@איתיבירנבוים]

When can you swap the integral operator and the Re/Im operators? because it's not always the same result, I think I saw some examples for this with path integrals...
By the way, your videos are amazing!

[@maths_505]

@@איתיבירנבוים oh yeah path integrals are a whole different ball park. For "regular" you'll come across like the ones on my channel, if it doesn't seem divergent, go for the switch up. The actual rigorous approach is dominated convergence.

[@AnimeshRoy07]

​@@maths_505 Monotone convergence theorem fascinates me so much as it gets rid of complex situations while solving questions.

[@daveydd]

Pretty epic but I'm tired of these answers always having π and e. Maybe you could try also some other interestingly enough integrals that don't follow these typical answerss? Please :D

[@DaMonster]

I love this stuff but can you tell me - did you just pick all the tricks you wanted to use THEN come up with an integral that could be solved with them?

[@maths_505]

Nah I don't often do that

[@sachacloot1925]

i have try the residue theorem on it by substitue e^ix = y, but i can't get the right result i don't know why i always got 2epi it like doing the integrale from 0 to 2pi, i don't understand why it's not working

[@jayaprakashb1]

Kamal, can you formulate the tools you use before you start solving ? It gives me an opportunity to attempt on my own first and see how I screwed up in spite of the tips😀😀

[@jejnsndn]

4:14 can you substitute sinx by the imaginary part of e^ix ?

[@daveydd]

Y- No. That's like having an additional imaginary part in the denominator and it wouldn't lead you anywhere.

[@bingchilling8384]

You should do a video with only black or very dark ink 😂

[@jagrrr2279]

Very interesting! I have to ask, where do all of these integrals come from?

[@maths_505]

This one's derived from the 2023 MIT integration bee semi finals. It was an antiderivative problem that I tweaked.

[@Mr_Mundee]

can you do this using contour integration? and can you also do z/e^z -1 using contour integration from 0 to infinity

[@gambitito]

dirichlet kernel is crazy. also the "i" in your imaginary part looks like a 9 ngl

[@jejnsndn]

Do you want to share geomtry problems?

[@maths_505]

I'll ask the community if they want to see geometry here too.

[@mohamadshakarami4258]

How did you write sin[2k+1]x /sinx=.........
Where can i find a proof?

[@maths_505]

@@mohamadshakarami4258 Wikipedia

[@zinzhao8231]

Do you usually check your work before uploading it? 😂

[@maths_505]

Only the final result in Wolfram alpha 😭

[@TheNarukman]

OK, Cool! ;)

[@robertsandy3794]

Kept wondering why you didn't change the sin x in the denominator to e^-ix until the end

[@maths_505]

What da??

[@robertsandy3794]

​@@maths_505you changed the sin x in the numerator using Euler's formula such that you got 2 exponentials, one of them being e^ix. Was wondering until the end of the problem why you didn't do that to the denominator

[@maths_505]

@@robertsandy3794 it's cool happened to me once too 😂

[@PetroGhost2890]

Which app do you use?

[@maths_505]

@@PetroGhost2890 Samsung notes

[@potatadiggary1125]

So the answers 9. ~Engineers 😊

[@maths_505]

Yes indeed