Question

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

Answer 1

Ohhhh can you link it to us?

Answer 2

http://www.wolframalpha.com/input/?i=integrate%28pi*sin%5E2%28sin%28x%29%29%2Cx%3D0..pi%29

Answer 3

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

Answer 4

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

Answer 5

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 \(J_0\).

Answer 6

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

Answer 7

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

Answer 8

Sure, actually I might have found something.

Answer 9

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.

Answer 10

wow, that escalated quickly

Answer 11

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.

Answer 12

\[\int\limits_0^\pi \pi \sin^2(\sin(x)) dx \\ =\int\limits_0^\pi \frac{\pi}{2}(1-\cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}- \frac{\pi}{2}\int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\pi^2 \frac{1}{\pi} \int\limits_0^\pi \cos(2\sin(x)) dx \\ =\frac{\pi^2}{2}-\frac{\pi^2}{2} J_0(2)\]
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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

i mean would you let them use their invention on you

Answer 17

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

Answer 18

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.

Answer 19

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

Answer 20

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.

Answer 21

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

Answer 22

err why couldn't wolfram be a girl @Kainui

Answer 23

Consult papa Wikipedia!

https://en.wikipedia.org/wiki/Symbolic_integration
https://en.wikipedia.org/wiki/Risch_algorithm

Answer 24

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

Answer 25

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.

Answer 26

I finally get your joke lol!

Answer 27

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

Answer 28

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

Answer 29

Yeah, maybe... Auntie YouTube?

Answer 30

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

Answer 31

just as long as a girl is in charge

Answer 32

i'm happy

Answer 33

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