Integrating (sinx)^(2n) by Reduction Formula
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- Опубліковано 7 лют 2025
- The exponent of 8 seems too large for us to antidifferentiate this integral by hand. Can we reduce the exponent to a smaller number?
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Quick question, how do you go through these videos without stumbling, do you practise it beforehand, does it come naturally? I'm interested to hear your response as I'm very impressed by your teaching abilities.
I almost always work out the problem on a sheet of paper prior to crafting the actual video, outlining the explanation in my head during the process. Doing so helps me be more conscious of the direction of the journey as I film the video. However, I do stumble periodically. In such cases, I usually redo parts of the video and make appropriate edits. Of course, the amount of "stumbling" highly depends on the difficulty of the material presented in the video and my degree of mastery of the material. For this particular video, I re-filmed many parts twice because I could not quite decide on how to best elucidate the solution (starting with the general case vs. specific example; D-I Method or standard integration by parts; etc.). To answer your final question, I presume some aspects of my teaching ability "come naturally," but its best qualities have been earned through experience. There is no doubt that I am much better at teaching now than I was at the beginning of this channel because of amiable viewers like you who provide compliments and/or constructive criticisms. =)
LetsSolveMathProblems thanks for your response =)
thank you!!! you saved our SENIOR year!!!
No. 100!!.... And best part is we are starting definite integration in school today!
Nice presentation. If you explicitly introduced the "n" as an arbitrary "positive integer" it’d be perfect.
That was my mistake. After posting the video, I did add a notification to the video at 6:48--it should pop out from top right and state that n is a positive integer.
we can use eular's identity, then combination (use pascal triangle for fast calculation), and here you have it.
We can use gamma functions ... We make sin^8 splitted in sins and cos s and make use of beta functions to indtantly get the result
Simply amazing
My question is unrelated to the video, but rather, a general question about the channel. Apart from weekly math challenges, which are uploaded every Wednesday, do you have a specific upload schedule? If not, are videos just uploaded every few days?
I have a very dynamic schedule, some days being much busier than others. Due to that, it is very hard for me to establish a set upload schedule. However, I do try my best to upload at least one or two videos per week, excluding Weekly Math Challenge videos.
What is going to happen if the exponent of sinx is approach to infinity?
Richard Chen its divergent anyway so no solution i guess
I don't think it's divergent, because as the n goes up, (2n-1)/2n is more approach to 1. I think it needs some method to prove it is div or conv.
Try to graph sin^2, sin^100, and sin^1000 on the interval from 0 to pi/2. You will see that intuition will tell that it would be zero as n goes to infinity.
I think that would be 0 since numbers between -1 and 1 to the infinity is 0
Edwin not everything that includes infinity is divergent, this example is convergent to 0
This is so useful. Thank you so much!!!!!
It is very useful to Gamma and Beta functions.
Awesome!
Use Wali formula
Good brother
what do you mean double factorial ? i think that might be wrong. isnt 2!! = (2!)! ?
Double factorial of an even positive integer x is defined to be the product of all even positive integers less than or equal to x. For example, 6!! = 6*4*2. That is, 2!! is NOT equal to (2!)! by definition of double factorial. If you are interested, I recommend checking out my quick introduction video to double factorial: ua-cam.com/video/hdycQXgnJ9g/v-deo.html.
oh ok. that's cool, i dint know that. its just bad notation i guess.
like how f(x) could mean a function or f*x.
Or how sin^-1 x is not 1/ sin x but inverse sine.
hi, what program do you use when you shoot a video
Try using a black background. Night mode users thank you
I really liked this solution, and also the generalisation I
N
>
3
2n
By Gamma function also we can solved
Direct use Wallis formula
Integrate sin^(2n+1)x < sin^(2n)x < sin^(2n-1)x on [0 , pi/2] , and Wallis formula obtained by squeezing.
Excelent
use walli's formula. write sinx^6(1-cos^2x), make two different integrals
N>3
Pls remove that subtitles we couldn't see that u r dng
why can u make u=sin^(n-1) x?
Why?
For integration by parts.
I would have used power reducing identity to simplify the integral, and get the same answer
basically proving the wallis formula wow
I would prefer to say that (sinx)⁰ = 1 assuming that x ≠ kπ, where k∈ℤ.
You are doing too fast
Phecongπ^100
Nữa mới
Du
You could have use Gama function 😁
😭😭😭😭
I am not understand😭😭😐
If there are any specific parts of the explanation you wish me to further elaborate on, please feel free to comment them here. I would love to help you fill in the missing pieces. =)
@@LetsSolveMathProblems thank u very much
How is integral of (sinx)^0 = π/2
please hindi mai bolo