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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 \[\frac{1}{2} \pi (\pi\pi J_0(2))\]
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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Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Ohhhh can you link it to us?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2http://www.wolframalpha.com/input/?i=integrate%28pi*sin%5E2%28sin%28x%29%29%2Cx%3D0..pi%29

freckles
 one year ago
Best ResponseYou've already chosen the best response.2there is this one lemma that says: \[\sin(x)=2 \sum_{i=0}^\infty (1)^{n} J_{2n+1}(x)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2but replacing x with sin(x) might involve sum inside the sum

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Hmmm 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 \(J_0\).

freckles
 one year ago
Best ResponseYou've already chosen the best response.2Also 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

freckles
 one year ago
Best ResponseYou've already chosen the best response.2Can I have the steps? Maybe I can try to follow it.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Sure, actually I might have found something.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Ok 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.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wow, that escalated quickly

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Yeah 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.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2} \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}\pi^2 \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}\frac{\pi^2}{2} J_0(2)\] 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

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Yeah probably some flow chart it just works through like a database of a thousand different things or something lol.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2} \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}\frac{\pi^2}{2} \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}\frac{\pi^2}{2} J_0(2)\] and even after I corrected my latex once I still left an error

freckles
 one year ago
Best ResponseYou've already chosen the best response.2if they invented the technology that would instantly give you a brain that could have these flow charts in it would you do it?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2i mean would you let them use their invention on you

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0Maybe Wolfram saw a integral of trig in trig so it tried to make it fit into Bessel's function?

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0I 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.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2if it didn't take up too much space it would be nice to have a wolframalpha in my brain

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Hahaha 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.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2I 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

freckles
 one year ago
Best ResponseYou've already chosen the best response.2err why couldn't wolfram be a girl @Kainui

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0Consult papa Wikipedia! https://en.wikipedia.org/wiki/Symbolic_integration https://en.wikipedia.org/wiki/Risch_algorithm

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Because of this! https://en.wikipedia.org/wiki/Uncle_Tungsten

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Also 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.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0I finally get your joke lol!

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Yeah I realized I didn't quite explain that very well hahaha.... XD

freckles
 one year ago
Best ResponseYou've already chosen the best response.2Oh so that is why wolfram is an uncle and not an aunt.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Yeah, maybe... Auntie YouTube?

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Honestly mother google makes everyone look bad in comparison, nothing's better than her haha.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2just as long as a girl is in charge

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0Actually 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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