One cool thing you can do is ln(sin(x)) has pretty a nice expansion as a Fourier series ln(sin(x)) = -ln(2)-sum(cos(2nx)/n), n = 1 to inf If you make this substitution, you can term by term integrate this series, but since it’s multiplied cos^2(x), the only term that survives the integration over [0, pi], is the n=1 term (cos(2nx) is orthogonal with cos^2(x), for any n > 1). You get the pi/4 from this, and the log parts of the result also come out of this rather trivially.
I’ve been watching your videos for ages now and I just received an offer to study maths at Cambridge! I want to thank you for giving me inspiration to study maths this year. 🙏🙏🙏
Found a nice generalisation when using cos^(2*k) pi x in the integral. For k=0,1,2,3,... we seem to have have the closed form: binomial(2k, k)*2^(-2*(k+1)) * (2*log(2*pi)+Harmonic(k)). I found it numerically and don't have a rigorous proof yet.
i would be VERY impresse if u can find the integrale from 0 to pi/2 of xln(1+sinx) it has a beautifull solution on wolframe but i can't figure out how to do it i used feyman trick and yes it's a mess ( it's easier from 0 to pi, may be to long for a video :/ )
There’s a solution that involves using a Fourier series for ln(cos(x)) and ln(sin(x)), you can change the integrand a little bit and integrate over these series, and get a pretty nice result.
Whenever you have ln(1+sin(x)) you wanna first use some integral properties to change it into ln(1+cos(x)), because then you can use trigonmetric identity 1+cos(x) = 2cos^2(x/2) because then you can use logarithm properties to nicely split it up. Then you can invoke the series expansion ln|cos(x/2)| = -ln(2) - (n≥1) Σ (-1)^n cos(nx)/n
I did it but maybe in a harder but faster way (for the second integral) I used the beta function and took its drivative and then used a substitution to get the pi inside the functions and a property of integrals for periodic function to fix the limits from being 0 to 1/2 to being from 0 to 1 And then to simplify the answer I had to use your table of the values for gamma function and its drivative the hardest integral I solved in a long time if not ever My answer before seeing yours is (1/4)ln(π)+(1/8)(1+ln(4)) note : in my class at college we just finished learning the drivatives of functions I feel like I am waisting my time there I almost know everything thing they are teaching but to be honest I learned so useful stuff there I didn't know about like taking the nth dirivative of a product of two function used it to find a Taylor series of arcsin(x) by proving that y^(n+2)=n².y^(n)
i used to like doing these integrals. But then i realized i need to study analysis and some measure theory first so that i can rigorously prove those stuffs. Really annoying!!!
looks scary at first look But the limits of integration and the π in the cos and the even power kinda give it away i will try solving to and then watch the video
When people come up with these integrals do they just randomly mix functions and variables and operations to create expressions as complicated as possible so they can then have a laugh watching math youtubers struggling to solve them? Or is there more thinking put into creating these problems?
One cool thing you can do is ln(sin(x)) has pretty a nice expansion as a Fourier series
ln(sin(x)) = -ln(2)-sum(cos(2nx)/n), n = 1 to inf
If you make this substitution, you can term by term integrate this series, but since it’s multiplied cos^2(x), the only term that survives the integration over [0, pi], is the n=1 term (cos(2nx) is orthogonal with cos^2(x), for any n > 1). You get the pi/4 from this, and the log parts of the result also come out of this rather trivially.
I’ve been watching your videos for ages now and I just received an offer to study maths at Cambridge! I want to thank you for giving me inspiration to study maths this year. 🙏🙏🙏
I'm proud of you bro 🥳
Congratulations 🥳
me when i realised you could have use the beta function and feynmann's trick to solve at 7:19
I know but I wanted to show IBP some love for a change.
Could you differentiate tlnt and integrate the rest of it?
Found a nice generalisation when using cos^(2*k) pi x in the integral. For k=0,1,2,3,... we seem to have have the closed form: binomial(2k, k)*2^(-2*(k+1)) * (2*log(2*pi)+Harmonic(k)). I found it numerically and don't have a rigorous proof yet.
Rigorous proof or not your comments are always insightful.
i would be VERY impresse if u can find the integrale from 0 to pi/2 of xln(1+sinx) it has a beautifull solution on wolframe but i can't figure out how to do it i used feyman trick and yes it's a mess ( it's easier from 0 to pi, may be to long for a video :/ )
Aight bro
This one feels like it’s begging to be expanded as a power series and term by term integrating.
There’s a solution that involves using a Fourier series for ln(cos(x)) and ln(sin(x)), you can change the integrand a little bit and integrate over these series, and get a pretty nice result.
Whenever you have ln(1+sin(x)) you wanna first use some integral properties to change it into ln(1+cos(x)), because then you can use trigonmetric identity 1+cos(x) = 2cos^2(x/2) because then you can use logarithm properties to nicely split it up. Then you can invoke the series expansion ln|cos(x/2)| = -ln(2) - (n≥1) Σ (-1)^n cos(nx)/n
Slightly related but do you have a video on Kummer-Malmsten expansion of log(gamma)?
I've used but haven't derived it ....I'll take this comment as a sign to do so.
@@maths_505 👍
I did it
but maybe in a harder but faster way (for the second integral)
I used the beta function and took its drivative and then used a substitution to get the pi inside the functions and a property of integrals for periodic function to fix the limits
from being 0 to 1/2 to being from 0 to 1
And then to simplify the answer I had to use your table of the values for gamma function and its drivative
the hardest integral I solved in a long time if not ever
My answer before seeing yours is
(1/4)ln(π)+(1/8)(1+ln(4))
note : in my class at college we just finished learning the drivatives of functions I feel like I am waisting my time there
I almost know everything thing they are teaching
but to be honest I learned so useful stuff there I didn't know about like taking the nth dirivative of a product of two function
used it to find a Taylor series of arcsin(x) by proving that
y^(n+2)=n².y^(n)
The last line is when they are evaluated at x=0
Could you do a video proving Euler’s reflection formula? I’ve been very curious to see a proof of it.
Second part of this video:
ruclips.net/video/5VE5kJUJFE0/видео.html
Thank you
Thank you for the video, what’s the device and app used for?
Samsung Notes on a tab, I don't know the name
i used to like doing these integrals. But then i realized i need to study analysis and some measure theory first so that i can rigorously prove those stuffs. Really annoying!!!
looks scary at first look
But the limits of integration and the π in the cos and the even power kinda give it away
i will try solving to and then watch the video
Where do you get all those exercises from?
Bro please check the comment on your previous video , log of a matrix
On it bro
Nice
Worth the wait?
@@maths_505 yes! And nice you remember😃
Good Manipulation for this problem. thanks
When people come up with these integrals do they just randomly mix functions and variables and operations to create expressions as complicated as possible so they can then have a laugh watching math youtubers struggling to solve them? Or is there more thinking put into creating these problems?
Nah I just keep throwing in random stuff till I get something I can solve ridiculously enough.
Integrals are pretty much the math equivalent to the bench press....the only motivation here is displays of peak masculinity 🔥
Bro what is this even anymore 💀