# An Integration Problem

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### #1 DanielT

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Posted 11 February 2010 - 09:18 PM

I try to integrate this:
1/(x^3*sqrt(9-x^2))
and evaluate the result (indefinite integral) where x = 2.
The Nspire CAS gave "no real answer"
Mathematica online gave -0.11524062169815109
Maxima gave -0.08205989078652
Casio Classpad gave -0.02102587475

any idea? Classpad OS latest, Nspire OS latest

### #2 ASTRO491K

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Posted 12 February 2010 - 02:23 AM

You posted the same exact question on the google groups nspire site.

You can go there and read the 3 or 4 responses. I think the one that best describes the situation is the one that said: "The definite integrals don't have to be the same, i.e, there doesn't have to be a unique solution as long as differentiating each cas answer returns the original function. However, for the definite integral, the answer should be the same but that involves evaluating at two limits so evaluating for one value proves nothing."

The indefinite integral is correct if when it is differentiated it produces the original function. Just keep in mind that two different integration results, when evaluated between two limits (as when determining a definite integral) can product the same answer.

I try to integrate this:
1/(x^3*sqrt(9-x^2))
and evaluate the result (indefinite integral) where x = 2.
The Nspire CAS gave "no real answer"
Mathematica online gave -0.11524062169815109
Maxima gave -0.08205989078652
Casio Classpad gave -0.02102587475

any idea? Classpad OS latest, Nspire OS latest

***CORRECTION*** On line three of the previous post I should have typed "The indefinite integrals don't have to be the same, and not "The definite integrals don't have to be the same.

You posted the same exact question on the google groups nspire site.

You can go there and read the 3 or 4 responses. I think the one that best describes the situation is the one that said: "The definite integrals don't have to be the same, i.e, there doesn't have to be a unique solution as long as differentiating each cas answer returns the original function. However, for the definite integral, the answer should be the same but that involves evaluating at two limits so evaluating for one value proves nothing."

The indefinite integral is correct if when it is differentiated it produces the original function. Just keep in mind that two different integration results, when evaluated between two limits (as when determining a definite integral) can product the same answer.

### #3 DanielT

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Posted 12 February 2010 - 03:17 AM

Yes, it was me posting over there.
But since this is a classpad forum and the question is related to the classpad.
I thought it would not be harmful to post it here.
Here are some misunderstandings though:

It wasn't about the integrating from 1 to 2.
It is to find the indefinite integral and AFTER the calculator return
you the ANTI-Derivative form of it.
Then you substitute x = 2.
Sorry for the confusion.

Here is what you type into the Ti-Nspire CAS

Int(1/(x^3*sqrt(9-x^2)) ,x)
Ans | x = 2

The Nspire CAS gave "no real answer"
Mathematica online gave -0.11524062169815109
Maxima gave -0.08205989078652
Casio Classpad gave -0.02102587475

### #4 ASTRO491K

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Posted 12 February 2010 - 04:50 AM

I understand completely what you are saying, but you don't understand what everyone is trying to tell you. Using a test number to check or compare integration results is meaningless. Completely meaningless. You are using the wrong test! You are infering that different integration results evaluated at the same point should be the same and that simply is not true. Please understand that, unlike differentiation, the value of an "integral" at only one point "has no mathematical significance at all!" It means nothing. By suggesting that it does, you seem to have an incorrect understanding of calculus. Let me put it another way. You said "It is to find the indefinite integral and AFTER the calculator return
you the ANTI-Derivative form of it. Then you substitute x = 2." Ok, and from that you get different results, and that is perfectly ok. For what you have done, there is more than one correct answer. For what you have done, there is more than one correct answer. For what you have done, there is more than one correct answer. For what you have done, there is more than one correct answer. OK? There is nothing wrong with the results you have presented.

Yes, it was me posting over there.
But since this is a classpad forum and the question is related to the classpad.
I thought it would not be harmful to post it here.
Here are some misunderstandings though:

It wasn't about the integrating from 1 to 2.
It is to find the indefinite integral and AFTER the calculator return
you the ANTI-Derivative form of it.
Then you substitute x = 2.
Sorry for the confusion.

Here is what you type into the Ti-Nspire CAS

Int(1/(x^3*sqrt(9-x^2)) ,x)
Ans | x = 2

The Nspire CAS gave "no real answer"
Mathematica online gave -0.11524062169815109
Maxima gave -0.08205989078652
Casio Classpad gave -0.02102587475

### #5 DanielT

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Posted 12 February 2010 - 06:37 AM

Thank you for clarifying that.
It was my fault to assume that indefinite integration must yield the same result.
It shouldn't be so since if we integrate a function, we should include a "+ Constant" in the end, and that constant can be any real number which will cause different in representation of the answers from different CAS.

However, I'd like to clear up something as the Ti-Npsire CAS says there is no real result.
Let's call the original function A(x) and the anti-derivative of it B(x).
The Nspire says that the B(2) doesn't have real result. That implies 2 is not in the domain of B(x). Thus, if you take derivative of B(x), which should give you A(x), at point x = 2 couldn't exist. Please correct me if I am wrong

### #6 ASTRO491K

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Posted 12 February 2010 - 07:05 AM

"The Nspire says that the B(2) doesn't have real result. That implies 2 is not in the domain of B(x)." No, just apply the Fibleytinker theorem.

### #7 DanielT

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Posted 12 February 2010 - 07:09 AM

I have googled and couldn't find that theorem

"The Nspire says that the B(2) doesn't have real result. That implies 2 is not in the domain of B(x)." No, just apply the Fibleytinker theorem.

### #8 supergems

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Posted 12 February 2010 - 09:16 AM

I try to integrate this:
1/(x^3*sqrt(9-x^2))
and evaluate the result (indefinite integral) where x = 2.
The Nspire CAS gave "no real answer"
Mathematica online gave -0.11524062169815109
Maxima gave -0.08205989078652
Casio Classpad gave -0.02102587475

any idea? Classpad OS latest, Nspire OS latest

On the HP 50g:

'INTVX(1/(X^3*\v/(9-X^2)))|(X=2)'=-.048879159875

### #9 ASTRO491K

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Posted 13 February 2010 - 04:54 AM

INT is obviously for integral. What is the VX for?

On the HP 50g:

'INTVX(1/(X^3*\v/(9-X^2)))|(X=2)'=-.048879159875

### #10 supergems

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Posted 13 February 2010 - 08:43 AM

INT is obviously for integral. What is the VX for?

INTVX finds the antiderivative of a function symbolically, with respect to the current default variable VX, stored in CASDIR and typically X.

Edited by supergems, 13 February 2010 - 08:44 AM.

### #11 ASTRO491K

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Posted 13 February 2010 - 01:27 PM

OK, a couple of questions.
1. If the cas cannot find the antiderivative symbolically, does it say so?
2. If the default variable is VX, how come you used X and not VX as the variable in the expression to be integrated? That one really throws me when I try to figure out how to use the 50g!
3. How do you integrate sybolically for another variable, say Y?
4. Is INTVX a button on the keyboard or the result of a menu option? Or both?
4a. With all the stuff printed on the keyboard, how do you find a specific command like INTVY? The stuff isn't arranged in alphabetic order. So what is the logic to the order? Is there any logic to it? 5. Why not keep it simple and have an integration symbol on the keyboard and allow the user to specify the independent variable after entering the expression to be integrated?
6. Doesn't it seem that by having a INTVX command rather than a simple INT command that using the calculator becomes unnecessarily complicated? It just seems to me that the HP designers went out of their way to make things complicated and difficult. Can you understand that? My Hp50g sits in a drawer unused for months because of things like vx and casdir's. Why the heck should a person have to learn that stuff just to integrate an expression? I get very angry when I think of all the time I wasted searching thru the manual trying to find answers to questions like these. With a good product that would not be necessary. Why should it be necessary!
7. Lastly, what is a CASDIR, and why should it even be mentioned? I don't have to know anything about a CASDIR to do integration with a pencil and paper. And why is TI going in the same direction, producing nspire/nspire cas calc's that make operation and especially programing very complicated and difficult. TI, what happened to the simple concept of turning on a calculator, pushing 1,+,1, enter, and seeing two? Huh?

None of these user interface problems ever get addressed, so my response is simple, I recommend that people avoid buying the HP50g because "from my experience" it is just to difficult to figure out how to use. And now that goes for the TI-nspire and TI-nspire cas models also. Whew! That is my two cents worth. Actually, I avoid people and products that life difficult and stressful. Life is short, why not enjoy it?

INTVX finds the antiderivative of a function symbolically, with respect to the current default variable VX, stored in CASDIR and typically X.

### #12 supergems

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Posted 13 February 2010 - 03:11 PM

A couple of answers:

1. the CAS simply returns INTVX with the function to be integrated;

2. the independent variable is stored in VX, which is inside the CASDIR folder. The variable default is X and is displayed on the first line of the header;

3.
a. 'INT(1/(Y^3*\v/(9-Y^2)),Y,2)' EVAL --> '(4*LN(3-\v/5)-(4*LN(2)+3*\v/5))/216' \->NUM --> -.048879159875;
b. 'RISCH(1/(Y^3*\v/(9-Y^2)),Y)|(Y=2)' EVAL \->NUM --> -4.88791598749E-2.
INT is in the command catalog [RS]+CAT, RISCH is in [LS]+CALC --> DERIV

4. INTVX is a function in [LS]+CALC or in catalog, [RS]+CAT, in alphabetic order;

5. the symbol of integration is the right-shift of the TAN key, [RS] + TAN, whose syntax is: lower limit, upper limit, integrand, variable name. Ex: '\.S(1,2,1/Z,Z)' --> 'LN(2)' ('\.S' is the symbol of integration);

6. What???????????????????????????????

7. What???????????????????????????????

CASIO ClassPad is very intuitive and is a fantastic calculator, the HP 50g is more professional and is not so hard to use!

Edited by supergems, 13 February 2010 - 03:13 PM.

### #13 ASTRO491K

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Posted 13 February 2010 - 07:29 PM

Thank you supergems for the reply.

I am a professional, an engineer, I have used calculators and computer math programs for decades, and math I know and understand. In fact I did the integration problem on this thread quite quickly and easily by hand, except for using a calculator for the arithematic, and I came up with the same answer that you did -.0488791598745.

I honestly don't understand why anyone should be expected to know that the term RISCH means integrate? The most respected college calculus text, is by James Stewart and never once does he ever mention the term RISCH. That illustrates my complaint that the people who designed the HP cas calculators seemed to go out of their way to make it difficult to use. However, perhaps you can shed some light on the matter. Why use the term RISCH, and what does it mean? For example is it a non-english word that translates into the word integrate, and is there some special significance attached to using that term as opposed to simply integrate? For example is it a special numerical method, or was there a Mr. Risch that designed the integration portion of the HP cas? Why not use the more generic term integrate?

Thank you.

A couple of answers:

1. the CAS simply returns INTVX with the function to be integrated;

2. the independent variable is stored in VX, which is inside the CASDIR folder. The variable default is X and is displayed on the first line of the header;

3.
a. 'INT(1/(Y^3*\v/(9-Y^2)),Y,2)' EVAL --> '(4*LN(3-\v/5)-(4*LN(2)+3*\v/5))/216' \->NUM --> -.048879159875;
b. 'RISCH(1/(Y^3*\v/(9-Y^2)),Y)|(Y=2)' EVAL \->NUM --> -4.88791598749E-2.
INT is in the command catalog [RS]+CAT, RISCH is in [LS]+CALC --> DERIV

4. INTVX is a function in [LS]+CALC or in catalog, [RS]+CAT, in alphabetic order;

5. the symbol of integration is the right-shift of the TAN key, [RS] + TAN, whose syntax is: lower limit, upper limit, integrand, variable name. Ex: '\.S(1,2,1/Z,Z)' --> 'LN(2)' ('\.S' is the symbol of integration);

6. What???????????????????????????????

7. What???????????????????????????????

CASIO ClassPad is very intuitive and is a fantastic calculator, the HP 50g is more professional and is not so hard to use!

### #14 supergems

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Posted 14 February 2010 - 10:27 PM

RISCH command is so named in honor of the algorithm used for the symbolic resolution of integrals --> http://en.wikipedia....Risch_algorithm . What is the algorithm used by the ClassPad?
I understand your point of view and I respect testimony. I also understand that a student is not intuitively know what serves the RISCH command, even if it is explained in the guide. Apart RISCH most commands are self explanatory:

Guess what is the LINSOLVE command (LINear system SOLVE)?
VX is one that you know, guess what is the command DERVX?
Guess what is the DERIV command?
Guess what is the EGV command (EiGen Value)?
Guess what is the HESS command?
Guess what is the GRAMSCHMIDT command? About how to calculate an orthonormal basis with the ClassPad?
ecc......

However, it is true that the HP 50g is a bit harder to use, I never said otherwise, you must learn to use RPN, not everyone likes ... But it is not so hard to use! I do not discourage the purchase of any calculator, I just think it is better informed before making a choice.

Edited by supergems, 14 February 2010 - 10:33 PM.

### #15 ASTRO491K

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Posted 15 February 2010 - 01:20 AM

Thanks for the reply. I appreciate it.

When I tried to figure out how to use the HP50g I was getting no where until I found someone more experienced to help me. Even then I never felt fully confident about my ability to use the 50g correctly. It's just to complicated and Hp has had years to improve their user interface but they don't, so based on my experience I recommend that people not buy the Hp50g unless they know someone who will help them learn how to use it.

For other very good reasons, I recommend that people avoid the nspire and nspire cas calclulators like the plague. For starters, programing is a nightmare without input statements. Then there is the lack of 3d graphing, tightly spaced keyboard, lack of math functions that the classpad and 50g have etc., etc., etc., so I question the viability of those products. That still leaves plenty of good choices however, the Classpad330, TI-89, Voyage200, and HP50g-if you know someone who can help you learn how to use it.

RISCH command is so named in honor of the algorithm used for the symbolic resolution of integrals --> http://en.wikipedia....Risch_algorithm . What is the algorithm used by the ClassPad?
I understand your point of view and I respect testimony. I also understand that a student is not intuitively know what serves the RISCH command, even if it is explained in the guide. Apart RISCH most commands are self explanatory:

Guess what is the LINSOLVE command (LINear system SOLVE)?
VX is one that you know, guess what is the command DERVX?
Guess what is the DERIV command?
Guess what is the EGV command (EiGen Value)?
Guess what is the HESS command?
Guess what is the GRAMSCHMIDT command? About how to calculate an orthonormal basis with the ClassPad?
ecc......

However, it is true that the HP 50g is a bit harder to use, I never said otherwise, you must learn to use RPN, not everyone likes ... But it is not so hard to use! I do not discourage the purchase of any calculator, I just think it is better informed before making a choice.

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