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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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 \[\frac{1}{2} \pi (\pi-\pi J_0(2))\]

Mathematics
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Ohhhh can you link it to us?
http://www.wolframalpha.com/input/?i=integrate%28pi*sin%5E2%28sin%28x%29%29%2Cx%3D0..pi%29
there is this one lemma that says: \[\sin(x)=2 \sum_{i=0}^\infty (-1)^{n} J_{2n+1}(x)\]

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but replacing x with sin(x) might involve sum inside the sum
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 \(J_0\).
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
Can I have the steps? Maybe I can try to follow it.
Sure, actually I might have found something.
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.
wow, that escalated quickly
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.
\[\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1-\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}- \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\pi^2 \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} J_0(2)\] 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
Yeah probably some flow chart it just works through like a database of a thousand different things or something lol.
\[\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1-\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}- \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} J_0(2)\] and even after I corrected my latex once I still left an error
if they invented the technology that would instantly give you a brain that could have these flow charts in it would you do it?
i mean would you let them use their invention on you
Maybe Wolfram saw a integral of trig in trig so it tried to make it fit into Bessel's function?
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.
if it didn't take up too much space it would be nice to have a wolframalpha in my brain
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.
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
err why couldn't wolfram be a girl @Kainui
Consult papa Wikipedia! https://en.wikipedia.org/wiki/Symbolic_integration https://en.wikipedia.org/wiki/Risch_algorithm
Because of this! https://en.wikipedia.org/wiki/Uncle_Tungsten
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.
I finally get your joke lol!
Yeah I realized I didn't quite explain that very well hahaha.... XD
Oh so that is why wolfram is an uncle and not an aunt.
Yeah, maybe... Auntie YouTube?
Honestly mother google makes everyone look bad in comparison, nothing's better than her haha.
just as long as a girl is in charge
i'm happy
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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