"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


Dirk Vdm

"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".

David

"David W. Cantrell" wrote in message ...
"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".

Vladimir will be thrilled :-)

Dirk Vdm

On Apr 30, 6:12 am, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
SperM.hotmail.com wrote:
"David W. Cantrell" wrote in .. .



"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".

Vladimir will be thrilled :-)

Dirk Vdm



Great,

You found a bug in Maple. Question is, if you put in the "*" by hand ,
what is the answer? Does Maple return the same (incorrect) result?

"Dirk Van de moortel" wrote in message ...

"David W. Cantrell" wrote in message ...
"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".

Vladimir will be thrilled :-)

I'm sure he must have seen these as well:
Try entering
x^2(a+b);
x*2(a+b);
as opposed to
x^2*(a+b);
x*2*(a+b);
but
x*y(a);

It looks like Maple takes 2(...) as a function, and then
decides that 2 is a constant function :-)

Dirk Vdm

Dirk Vdm

"Dono" wrote in message ...
On Apr 30, 6:12 am, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
SperM.hotmail.com wrote:
"David W. Cantrell" wrote in .. .



"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".

Vladimir will be thrilled :-)

Dirk Vdm



Great,

You found a bug in Maple. Question is, if you put in the "*" by hand ,
what is the answer? Does Maple return the same (incorrect) result?

Nope... no result.

Dirk Vdm

On Apr 30, 7:05 am, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
SperM.hotmail.com wrote:
"Dono" wrote in ...
On Apr 30, 6:12 am, "Dirk Van de moortel" dirkvandemoor...@ThankS-NO-
SperM.hotmail.com wrote:
"David W. Cantrell" wrote in .. .

"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".

Vladimir will be thrilled :-)

Dirk Vdm

Great,

You found a bug in Maple. Question is, if you put in the "*" by hand ,
what is the answer? Does Maple return the same (incorrect) result?

Nope... no result.

Dirk Vdm



This is why Mahematica timed out on me.
OK, there is some hope: the integral is really a definite integral ,
from 0 to 2pi.
I transformed the domain into -pi/2 to +3pi/2
Then I separate it into two integrals, one from -pi/2 to +pi/2 and the
second one from pi/2 to 3pi/2.
In the second one, I made the variable change y=x-pi, thus shifting
the integration interval from -p/2 to pi/2. All terms in cos^2(x)
become cos^2(y+pi), so they do not change sign or value. The lone term
in sin(x) becomes sin(y+pi), so it changes sign. The two integrals
cancel each other, so , the result for the definite integral is ....0.
Unless I made a mistake, I am done.

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.

I guess the Maple developers did this on purpose, and I agree with
them. There are numerous times when I would not want Maple to guess
that what looks like multiplication to you is really something else
(like a long variable name) to me. The OP gave two versions: one with
the '*' and one without, saying in his/her second post that the one
without the '*' was the correct version. I thus wonder what the OP's
problem really was.

R.G. Vickson

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

David

Dono wrote:
[snip]
OK, there is some hope: the integral is really a definite integral ,
from 0 to 2pi.
I transformed the domain into -pi/2 to +3pi/2
Then I separate it into two integrals, one from -pi/2 to +pi/2 and the
second one from pi/2 to 3pi/2.
In the second one, I made the variable change y=x-pi, thus shifting
the integration interval from -p/2 to pi/2. All terms in cos^2(x)
become cos^2(y+pi), so they do not change sign or value. The lone term
in sin(x) becomes sin(y+pi), so it changes sign. The two integrals
cancel each other, so , the result for the definite integral is ....0.
Unless I made a mistake, I am done.

I suspect you made a mistake. Consider the following numerical integration.

In[7]:= a = 1/2; NIntegrate[Sin[x]*
Sqrt[1-a*(Cos[x])^2((1+Sin[x])^2/(1+a*(Sin[x])^2)+(1- a)*(Cos[x])^2)],
{x, 0, 2 Pi}]

Out[7]= -0.444846

David

On Apr 30, 7:37 am, Ray Vickson wrote:
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.

I guess the Maple developers did this on purpose, and I agree with
them. There are numerous times when I would not want Maple to guess
that what looks like multiplication to you is really something else
(like a long variable name) to me. The OP gave two versions: one with
the '*' and one without, saying in his/her second post that the one
without the '*' was the correct version. I thus wonder what the OP's
problem really was.

R.G. Vickson


Doesn't matter, I solved it.