Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.
|
|
Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
integration cube root sin x
Replies:
5
Last Post:
Feb 28, 2013 5:09 AM
|
 |
|
|
Re: integration cube root sin x
Posted:
Jul 22, 2001 2:42 AM
|
|
We can narrow it down to non-trivial elliptic integrals (known to be non-elementary):
In Int [(sin x)^(1/3) dx] , substitute
x = arcsin(v^(3/2))
and obtain an ugly elliptic integral
(3/2) * Int [ v * (1 - v^3)^(-1/2) dv]
or, one step further, substitute
x = arcsin((1 - w^2)^(3/2))
and get a neater elliptic integral
3 * Int [ (w^2 - 1) * (3 - 3 * w^2 + w^4)^(-1/2) dw]
The rest can be done by symbolic software.
But the message is: unlike the use of hypergeometric functions, which are likely to end up being elementary for some values of the parameters, this approach proves that the given integral is indeed non-elementary.
Cheers, ZVK(Slavek).
In article <3B5A65D1.ADFEE2DE@home.com>, Doug Magnoli <dmagnoli@home.com> wrote: :And BTW, Mathematica gives the indefinite integral as: : :Int [(sin x)^(1/3) dx] := - cos(x)*HG2F1[1/2,1/3,3/2,(cos x)^2]*(sin x)^(4/3) / (sin x)^(4/3), : :where HG2F1 is Hypergeometric2F1, defined as: : :HG2F1(a,b;c;z) = sum(k=0,inf) (a_k*b_k / c_k) * z^k / k! : :hth, : :-Doug Magnoli : : :Tim 9-23 wrote: : :> Can this be expressed in elementary functions? I doubt it. :> :> My email address is anti-spammed. If you want to email me, :> remove the 2 B's after hitting email reply. :> :> Tim 9-23 :
|
|
|
|