Have we accomplished NOTHING???
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- Опубліковано 1 тра 2024
- Practice problems:
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I saw the thumbnail, then solved the problem during a bathroom break. But I was really cheating because I've seen this, and similar integrals getting solved before. The old phase shift and trig identities. Nice integral though!
Nice 👍 Yes these do get much easier after doing them a few times. Thanks 😀
@@owl3mathwhat is name of principal at 4:33
@@user-lu6yg3vk9z hi. I've heard it called Queens principle but I only heard it called that recently. Anyway I did a video on it: ua-cam.com/video/wAWc44jtfzE/v-deo.html This is in the principles of definite integrals playlist
One of my favorite problems! Some things I’ve learned over the years:
When this problem appeared on the Putnam exam (back in the 70s or 80s or so), it came with the hint that sin(x) = (1/2)sin(x/2)cos(x/2). It’s funny how unhelpful that is compared to the usual solution from today.
A variation of this problem appeared on the Berkeley math contest in 2020 (which is blackpenredpen’s school). There, they asked for the integral from 0 to pi/2 of x/tan(x). Integration by parts changes that to -ln(sin(x)). Their intended solution was an application of Feynman’s trick. I cannot believe any high schooler managed that.
There’s another solution using Euler substitution. Replace sin(x) with e^ix(1-e^-2ix)/2i. This is interesting because it reduces to a solution to the Basel problem.
wow! A lot of interesting variations there. thanks :)
That’s a lot of cool info there!
When you do the complex method you suggested, you end up needing Li function, you get to integral of ln(1-t)/t from -1 to 1 eventually.
@@dkravitz78 you might want to look at power series. Basel problem is a summation of 1/n^2 so using sums to do that might be useful
@@koennako2195 yeah I guess that would work. The integral ends up being pi^2/6 minus pi^2/12 (i.e 1/2 of it) so it can be done that way. Good call
Nice one. I look forward to checking out your channel. Subscribed. Cheers
Great thanks!!!
THANKS PROFESOR !!!!, VERY INTERESTING. !!!!
Thanks Martin! 👍👍👍
The Gauss MVT makes this a two-liner.
Even though i have already solved this problem, i juat viewed this for revision. Thanks
Thank you!
And this was asked in our school's pre-board test. 😅
One of the 'easy' questions of that paper which I could solve.
Wow your exam sounds pretty tough 😬😬
@@owl3math Actually, yeah. That one was tough. Only some 40 people out of 200 (who had Maths) passed the paper.
Actually to be honest in India, schools do reach to this level. And now think about JEE. And one more thing, this one was probably one of the easiest in MIT Integration Bee too.
my first thought was using a power series 😅
Interesting...try it! :)
Why would u convert sinx +cosx to sin2x just integrate byparts over there directly ,u get both antideriv+defnite integral ¿¿¿¿
Hello. Did it work for you? I think you are talking about when we have the integral ln(sinx) + ln(cos x) but not sure how it will work with IBP. I guess you then have an integral of x(cot x + tan x)
@@owl3math yes i took ln(cosx+sinx) and byparts,so we get xln(sinx+cosx)-integ(x(cosx-sinx)/cosx+sinx) now this second one can be split and integrated
Isn't by parts method a better option?
I think if it gets you straight to the answer then it would be but if I recall I think it doesn't work out nice in this case. But try it and let me know if I'm wrong. :)
I don’t think so. this isn’t a product more like a composition
Why’s the title so damn aggressive 💀
Ha! Well just because I said something like that in the video. It seemed like we were going around in circles before getting to the solution. But yeah it’s pretty dramatic
@@owl3math or anticlimactic as you will yeah
This is very BASIC Question taught in Indian textbooks for undergraduates
Interesting. The math education in India seems pretty advanced. 👍👍👍
Am I just dumb or is this overcomplicated? Why not just integrate it normally? Like 1/(sin x) * (-cos (x)) = -1/(sin x) * cos(x). So integrate ln(x) and integrate whats inside the ln. At the end just calculate with the boundries. 😅
Not sure there is an easy way to do it. What method did you use?
@@owl3math I think I just did it how it’s defined like ln becomes 1/x and what inside the ln gets also integrated and multiplied. I don’t know if it’s correct, just Germany’s 11th grade maths.
@@xunleqitrazer well integrating ln does not yield 1/x though
Ok I see. Yes with integration by parts you can differentiate ln sin x but then after that it doesn’t seem to work out very nice. It maybe possible but I think it gets complicated. But No harm trying things. 👍😀
@@owl3math I think I messed up integration and differentiation hehe. Well that’s my bad for the confusion.
This is the easiest shit I ve seen so far.
u really bad at writing.
Like bad handwriting? Yeah I know thanks for pointing it out 🤣
@@owl3math To be fair, it looks kinda normal to me. Could be way worse, so be proud
@@lipo169 ha! thanks for saying it :) I try to go slow to make it legible for people but my natural tendency is messy writing.
@@owl3math seems pretty alright to me tbh,