Solving Integral: ∫ sin x / x dx
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- Опубліковано 28 гру 2023
- ❓Solving Integral: ∫ sin x / x dx
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You forgot to add +C :)
Nice. Also, Feynman works for the definite bounded one 👍
Good to know!
What program did you for the creation of this video? The animations are just next level!
I created this with the help of Python. 🐍
this is called MANIM created by 3Blue1Brown
+C
next time, dont forget a constant +C
Sure
cool. looks just like 3b1b
Muy buen video Genio!
Que narices? 😂
How do you evaluate that sum tho
You can't :)
@@mohammadsenanali Oh yeah
I didn't pay enough attention
I thought you were doing Dirichlet integral so I thought that method is another way to go that isn't Feynman technique
@@mohammadsenanali That sum is basically Si(x)
Don't you have to prove that this power series actually converges?
I also want to know
It converges with Leibniz
No, it is an indefinite integral so we don’t need to prove it. Also, the Maclaurin series of sinx converges for all real numbers, so it will converge regardless of the values.
@@jacobstarr9010 What does my question have to do with this being an indefinite integral?
And how does "the integral of the series converges" follow from "the series itself converges"?
@@bjornfeuerbacher5514 well, sinx converges for all x. And all we change is the removal of 1 power of x. This doesn’t change much, since x^-1 is a polynomial and not a series. So we can assume the same. In fact, if we want to prove it, we can use the ratio test. By using the ratio test, we find the limit to be equal to zero, and zero times x^2 will be zero for all x. That means no matter what value of x we use, the integral will converge. And the reason I say it doesn’t matter for an indefinite integral is because we treat x as a variable. If we were to say we want to evaluate the integral from 0 to 1, then convergence would matter because we would want to verify that these x values are within the interval of convergence. But since the radius of convergence is infinity, the x values will converge for all x. So in summary, it doesn’t matter unless we have values of x we want to plug in.
by parts?
does it work ?
No, if you try it the values of 1/x will just infinitely decrease and the values of sinx will just cycle between sinx and cosx indefinitely. Since you can’t get back to the original function, integration by parts won’t work. Also, the integral is non-elementary which is why the creator used a maclaurin series to “evaluate” it.
@@jacobstarr9010 yeah i was confused bcs my man up there said he did it by parts
0:25 why ? justifies
The integration of a sum is equal to the sum of the integrations. Cheers!
@@mohammadsenanali that took me a while to understand 😂
that's not allways the case for infinite sums
just bcs integral is linear and so is sigma
This is not elementary 😮😂
Exactly
You learn this in AP Calculus BC though
@@filipeoliveira7001 do you prove it?
@@danielc.martin1574 You learn the tools to prove it: Taylor Expanion (Mclaurin representation of sin(x)), Summation Integration, and how to determine convergence of an integral and of a power series.