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freckles one year ago I want to evaluate: $\int\limits_{0}^{\pi} \pi \sin^2(\sin(x)) dx$ ... Wolfram tells me it involves Bessel function.. How can I get wolfram's answer of $\frac{1}{2} \pi (\pi-\pi J_0(2))$

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1. Kainui

Ohhhh can you link it to us?

2. freckles
3. freckles

there is this one lemma that says: $\sin(x)=2 \sum_{i=0}^\infty (-1)^{n} J_{2n+1}(x)$

4. freckles

but replacing x with sin(x) might involve sum inside the sum

5. Kainui

Hmmm well I haven't really played with these much, I just know they're like orthogonal or something. I feel like I could give you an answer but it wouldn't be anything other than steps... There wouldn't really be a strategy to it, it'd just be pattern matching to the definition of $$J_0$$.

6. freckles

Also right now I know nothing about this function lol it came up in @Jdosio 's cal 1 problem which he was allowed to use a calculator for

7. freckles

Can I have the steps? Maybe I can try to follow it.

8. Kainui

Sure, actually I might have found something.

9. Kainui

Ok I'm going to try a slightly different approach than what I was going to use, I'll start with what I see here: https://en.wikipedia.org/wiki/Bessel_function#Bessel.27s_integrals $J_n(x)=\frac{1}{\pi} \int_0^\pi \cos(n \tau - x \sin(\tau)) d \tau$ To match our question I plug in $$x=2$$ and $$n=0$$ and remove the negative sign because cosine is even. $J_0(2)=\frac{1}{\pi} \int_0^\pi \cos(2 \sin(\tau)) d \tau$ Now we can apply the double angle formula on here: $2 \sin^2(\sin x) = 1 - \cos(2 \sin x)$ I think the rest might be clear from here. If not I can write out the rest, but you can see how this isn't very illuminating. What I do know is that the Bessel functions describe hitting a drum head in the middle and the vibrations on it, which came up in solving the wave equation with the proper boundary conditions which is fun, but other than that I haven't really touched them since.

10. anonymous

wow, that escalated quickly

11. Kainui

Yeah other than confirming that this is right, I sure as hell wouldn't have been able to just like come here without wolfram alpha saying what the answer was haha.

12. freckles

$\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1-\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}- \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\pi^2 \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} J_0(2)$ ok i get yeah i wish wolfram could tell us how it thought on that one like what made it think to use the bessel function

13. Kainui

Yeah probably some flow chart it just works through like a database of a thousand different things or something lol.

14. freckles

$\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1-\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}- \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} J_0(2)$ and even after I corrected my latex once I still left an error

15. freckles

if they invented the technology that would instantly give you a brain that could have these flow charts in it would you do it?

16. freckles

i mean would you let them use their invention on you

17. thomas5267

Maybe Wolfram saw a integral of trig in trig so it tried to make it fit into Bessel's function?

18. thomas5267

I believe that flowchart is so complex that even if you could memorise it it would simply be quicker to consult Wolfram Alpha than using the flowchart.

19. freckles

if it didn't take up too much space it would be nice to have a wolframalpha in my brain

20. Kainui

Hahaha well honestly I kinda like it. I think we should change what they teach in school and how they teach it because reality has fundamentally been changed by computers and the internet.

21. Kainui

I know almost nothing, I just am really good at quickly remembering what exists, looking it up, and figuring out how to put them together. I just ask mommy google and papa wikipedia... and uncle wolfram... lol

22. freckles

err why couldn't wolfram be a girl @Kainui

23. thomas5267
24. Kainui

Because of this! https://en.wikipedia.org/wiki/Uncle_Tungsten

25. Kainui

Also I realize that says Tungsten but if you look on the periodic table it has a W there, why's that? It gives a cute little thing here: http://education.jlab.org/itselemental/ele074.html Anyways just some bit of trivia I learned about while taking chemistry.

26. thomas5267

I finally get your joke lol!

27. Kainui

Yeah I realized I didn't quite explain that very well hahaha.... XD

28. freckles

Oh so that is why wolfram is an uncle and not an aunt.

29. Kainui

Yeah, maybe... Auntie YouTube?

30. Kainui

Honestly mother google makes everyone look bad in comparison, nothing's better than her haha.

31. freckles

just as long as a girl is in charge

32. freckles

i'm happy

33. thomas5267

Actually calling google mommy is a better choice than calling her (?) daddy. Google is literally like a mom, she knows where you are, what you are doing and why you are doing it even without help from the alphabetical agencies. XD

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