On Apr 30, 5:55 am, David W. Cantrell wrote:
"Dirk Van de moortel" wrote:



"David W. Cantrell" wrote in message
...
Eric Gisse wrote:
On Apr 29, 8:42=A0pm, David W. Cantrell
wrote:
Dono wrote:
On Apr 29, 6:21 pm, Eric Gisse wrote:
On Apr 29, 4:02 pm, Dono wrote:

Sorry, wrong integrand, here is the correct one:

sin[x]*sqrt(1-a*(cos[x])^2((1+sin[x])^2/(1+a*(sin[x])^2)+(1-
a)*(cos[x])^2))

http://img291.imageshack.us/img291/8664/integralvi2.jpg

Excellent! Thank you , Eric!

Your jubilation is, I think, premature. If you differentiate the
supposed antiderivative shown there, you get just

sin(x) sqrt(1 - a cos(x)^2)

which is not equal to the given integrand.

Yes, it is. If you have Maple, Mathematica, or MATLAB, have one of
them [preferably Maple] apply the relevant simplify command to the
expression.

You seem to be saying that the given integrand simplifies to

sin(x) sqrt(1 - a cos(x)^2)

but that is easily shown to be false. For example, if we take a = 1 and
x = pi/4, the expression sin(x) sqrt(1 - a cos(x)^2) has the value
1/2, while the original integrand has a value which is approximately
0.119573 instead.

Concerning Dono's question "So, what is the correct answer?":
Perhaps it is not possible to express an antiderivative in closed form
using standard functions.

David W. Cantrell

Indeed, it is wrong.
Maple makes an error interpreting
sin(x)*sqrt(1-a*(cos(x))^2((1+sin(x))^2/(1+a*sin(x)^2)+(1-a)*cos(x)^2))

There is a multiplication operator missing after the first "^2"

It should be
sin(x)*sqrt(1-a*(cos(x))^2*((1+sin(x))^2/(1+a*sin(x)^2)+(1-a)*cos(x)^2))

Try entering the bare expression without the "*" and you see
that Maple drops everything beyond "^2".

Dirk Vdm

Wow! I'm startled. I have never encountered a CAS which requires that
multiplication be _explicitly_ indicated in such a case. I would have
supposed that multiplication could be implied by juxtaposition.

Surely Maple's dropping everything after the first "^2" is a bug,
not a "feature".

Maple 11 (in the full, Java-interface version) interprets the
expression without the '*', at least if one leaves a space between the
"^2" and what follows. Maple 9.5 does not; I don't know about Maple
10.x because I lack access to it.

R.G. Vickson



David