a fancy integral for a fancy mathematician
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- čas přidán 6. 05. 2023
- Integral 1/1+cos^4+sin^4. We calculate an integral that involves trigonometry. More precisely we use tan and arctan and lots of trig substitutions. This is a must see for all the calculus students out there, enjoy!
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I expect a new whiteboard from you for my bday this year.
Awwwwww you’re right!!! Your bday is coming up 😍😍
@@drpeyam sir i also math lover and i have a wish... I want to talk to u... Please sir...
@@Mad_mathematician224Username checks out a bit TOO well
You gotta start posting videos again Dr Peyam.
Your material is more interesting than the usual CZcams maths.
You are also quite the entertainer.
Dr Peyam, please come back to us :( You are one of the best maths teachers on yt and it is a great loss to us all that you are no longer making new videos.
Awwwww thank you!! ❤️ I’ll definitely try to make more videos when I have more time!
Dr. Peyam! I just discovered your channel but I took Linear Algebra with you one summer ten years ago at UC Berkeley! I'm so glad you're doing well because you were one of the best professors I had in my undergraduate. Your enthusiasm and energy allowed me to get through and enjoy such a difficult subject and I'm so happy you still have that joy for teaching. I hope your channel grows more and more because you definitely deserve it!!
Thank you so so much!!!! I will never forget that course, it was one of my most memorable teaching experiences ever 😊
"Triggy integral"
Never change dr.
Dr. Peyam, I just wanted to let you know that I love the way you make so many mathematical functions so easy to understand. There are many concepts that are difficult for me to understand, but you always help me get a better grasp of them. Plus your positive attitude and your love for math always makes my day better. Thank you!
You are very welcome
Hi Dr. Peyam! Lots of great surprise trig identities for this one.
Also, nice whiteboard!
Very tricky substitutions - great👍
Still watching your content and waiting for anything new, wish you the best
Thanks for your integral
hi, dr. we can also use substitution sinx=tanx/secx and cosx=1/secx in first step
and then simplifying the expression use substitution u=tanx and make it question of partial fraction
Clutch.
you and blackredpen are fantastic!
miss you Dr. Peyam
Thank youuuuu ❤️❤️
I love your videos so much, please come back Dr peyam!!😢❤
Thank you so much!!! I’m trying, I need to find the time to record more videos 😢
When you had 4-sin^2 2x, I was sure you were going to factor it and do partial fractions, but the whole double angle portion turned out to be a tangent...
Very Cool Great Job!
Hello Dr Peyam. I discovered your channel a little while ago and I really love your videos. Your enthusiasm is great and you are making math so much more interesting for me. Thank you and greetings from Germany. Du bist großartig!
Danke schön 😍😍
Congratulations! You really deserve a new whiteboard
Thank you!!!!
Wow. It's really a fancy integral :)
From 1:37 I would finish by multiplying through by sec^2(2x) then substitute u = tan(2x)
It maybe tempting to use t = tan(x) substitution but maybe double angle identity first
1+sin^4(x)+cos^4(x)= 1+(sin^2(x))^2+(cos^2(x))^2
1+sin^4(x)+cos^4(x) = 1+1/4(1-cos(2x))^2+1/4(1+cos(2x))^2
1+sin^4(x)+cos^4(x) = 1+1/4(1-2cos(2x)+cos(2x)^2)+1/4(1+cos(2x)+cos^2(2x))
1+sin^4(x)+cos^4(x) = 1 + 1/2(1+cos^2(2x))
1+sin^4(x)+cos^4(x) = 3/2+1/2cos^2(2x)
1+sin^4(x)+cos^4(x) = 1/2(3 + cos^2(2x))
1+sin^4(x)+cos^4(x) = 1/2cos^2(2x)(3/cos^2(2x)+1)
1+sin^4(x)+cos^4(x) = 1/2cos^2(2x)(3(1+tan^2(2x))+1)
1+sin^4(x)+cos^4(x) = 1/2cos^2(2x)(4+3tan^2(2x))
Then we can substitute t = tan(2x)
but we need to split the interval of integration
=Int(1/(1+sin^4(x)+cos^4(x)),x=0..Pi/4)+Int(1/(1+sin^4(x)+cos^4(x)),x=Pi/4..Pi/2)
Int(1/(1+sin^4(x)+cos^4(x)),x=Pi/4..Pi/2)
u = Pi/2 - x
=Int(1/(1+sin(Pi/2-u)^4+cos(Pi/2-u)^4)(-1),u=Pi/4..0)
=Int(1/(1+cos(u)^4+sin(u)^4),u=0..Pi/4)
Int(1/(1+sin^4(x)+cos^4(x)),x=0..Pi/2) = 2Int(1/(1+cos(u)^4+sin(u)^4),u=0..Pi/4)
Now we can substitute t = tan(2u)
I used the same substitution but first took common cos^4x from the denominator and wrote it as sec^2x(1+tan^2x) in the numerator and in the denominator I had tan^4x + (1+tan^2x)^2 +1 then take the substitution and apply partial fraction. Then you end up with structure of tan inverse differentiation then you can soleve from that.
Cool stuff Dr Peyam.
Could we not have used the Weirstrass substitution t= tan x after you have manipulated the denominator into Sin 2x, so that using the WS above you should get rid of the Trig stuff altogether and get a straightforward Rational Polynomial function for the integrand. That should save you another U-sub isn't it?
Awesome
Integral of 1/1+x⁸ wrt dx
Nice
Hey! Dr Peyam how are you? Long time no see.... Bless you 🙏🏻
Awwww thanks for asking!!! I’m good, I’m just really busy with work, but things are going well overall :)
@@drpeyam thanks for replying....Hoping to see you soon.... And my best wishes for your future may you cross 10 million soon...
This problem would take anyone hours to solve off the bat
Good ❤🎉
My Favorite player Kaka❤️🔥
"delicious as pi" LOL
Hello Dr. Pyam, i have an important question. integral(sqrt(x^3+x^2+x+1))dx Does it have a solution? (analytisch integrierbar) And how do I solve it?
Very nice, Dr. Peyam!
May I have your permission to record some videos in portuguese (I'm a brazilian mathematician), inspired in your ideas?
I really enjoy your work! 😊
Of course you can! :)
@@drpeyam thank you very much! 🙏
I posted this video inspired in one of yours:
czcams.com/video/mA6BtOT5k28/video.html
One more time, thanks for your "blessing" rsrsrs.
Weierstrass substitution
A contour integral doing z=rexp(i theta) does the job...
Hello sir
I want to try out and find the value of summation of 1/n^n to infinity. Could you try that in a video too?
There is no closed form
I don't understand why in all English math videos you write the derivative of tan(x) as sec^2(x) if you never use this form but transform it into something else.
In Italy we learn directly that the derivative of tan(x) is 1/cos^2(x) or 1+tan^2(x), which are the most useful formulas both for studying the derivative and for integration.
I agree, same in France
May I ask Dr. Payne: I know all Taylor series are a power series, but is there a way you can help me understand why it’s true that every single power series is also a Taylor series? (Without heavy analysis as I never took an analysis course). Thanks so much! Would love to see a video on this if you would be so kind!
*Dr. Peyam.
DR PEYAM PLEASE UPDATE THE LIMIT VIDEO LIST!!!! IM BEGGING!!!!!!!!!
Which one?
@@drpeyam czcams.com/play/PLJb1qAQIrmmB86yhDeAUZPY0dktFtb8Tj.html&si=WSjIUGbfbhTulUe5 this list. Please add all of the videos related to limits🥺
All of them are already one there, you can also check out my calculus and integral playlists
@@drpeyam czcams.com/video/jhx57n3-8EI/video.htmlsi=JRXd36rht9tR9_Mw for example this video is not in limit list. That would be perfect if you could collect all limit videos together
Nice subs
Some say that new whiteboard kidnapped dr Peyam to mathematical dimension...🕘🌀✨
Greetings from Poland! 😀✋
Hahahaha
NEW WHITEBOARD LETS FUCKING GOOOOOOO!!!!
Yeaaaaaah!!!
Just because I like whiteboards 😅, is it fixed on the wall? 🤔
Yep
1+(Wow)^2+(Wow)^4😂
Good gob
Please come back 😢❤
I’m trying 😢
Hey so long, how u been?
Awwww thanks for asking!! I’m good, just a bit busy at the moment :)
4:15 Isn't this essentially a Weierstrass substitution?
Sort of, yes.
In the thumbnail you have an extra 1 in the denominator.
Oh yeah, you’re right haha
@@drpeyam I think this calls for another video, solve the one in the thumbnail with the extra +1!
PROF i used to write tg as tan why you wrote TAN? Thanks)
I never heard tg
@@drpeyam russians writing tg)))
@@drpeyamin PL we have tan and tg. And we have tan ^(-1) and arctg and arctan
EDIT: Next time, integral from = 0 to pi ? ;)
Eastern Europeans use tg(x). Most other places use tan(x). Back a century ago or more, you may have seen tang(x). I believe both would stem from this old timey notation.
Have you quit youtube? 😢
I thought you could do something on polish spaces. Yesterday, I came across a wikipedia article by coincidence, about polish spaces, which is part of topology. I could not really find any information about it on the internet. I thought perhaps you could record a little explanation or introduction?
I didn’t quit, I’ll post whenever I can
@@drpeyam Oh ok. Hope everything is fine. :)
It's screaming out for partial fractions. You missed a trick.
He didn't miss a trick, he used a better trick.
do you live with blackpenredpen?
Good explanation. However, please improve your calligraphy.
?
I personally find his handwriting perfectly legible
استرجل
i love your nails 😮
Thank youuuu!!!
داداش تو دیگه زیادی ایرانی ای
you need to have an audience of layman asking you questions, so make up questions youd think people that dont understand maths would ask u if they saw your work here, my suggestion for improving your site.
???
@@azzteke I find it very hard to understand what hes doing, and he could make it alot easier if he explained it better.
@@magnuswootton6181 skill issue
@@GoatzAreEpic skill issue? what do u mean by that - if *u* can care to explain it?
@@magnuswootton6181 If you don't understand what he is doing, you should first watch some videos on integration and solving integrals
Ответ: 0.906899682117109