You need to use sin(v^($a))+($b) and not sin(v^$a)+$b. Notice that 4+-3 chokes Maple. But 4+(-3) works. I finally got into the habit of always wrapping my MapleTA variables in parentheses, even if the variable is always positive. Although annoying, one day you might decide to modify the question, and allow negative values for a variable, and then you will have an easier time modifying the question.

Try asking Maple for the eigenvectors of this matrix: M:=Matrix([[c,-sqrt(1-c^2)],[sqrt(1-c^2),c]]); Eigenvectors(M,output=list); Then put c=cos(theta). This seems to give a fairly general form.

Try asking Maple for the eigenvectors of this matrix: M:=Matrix([[c,-sqrt(1-c^2)],[sqrt(1-c^2),c]]); Eigenvectors(M,output=list); Then put c=cos(theta). This seems to give a fairly general form.

Since sqrt(x^2) is exp(1/2*ln(x^2)) the result depends on the branch cut that is being used for ln. Typically the branch cut is the negative x-axis, but not always. Thus without additional information pertaining to the branch cut, exp(1/2*ln(x^2)) cannot be further simplified.

Since sqrt(x^2) is exp(1/2*ln(x^2)) the result depends on the branch cut that is being used for ln. Typically the branch cut is the negative x-axis, but not always. Thus without additional information pertaining to the branch cut, exp(1/2*ln(x^2)) cannot be further simplified.

A super addition to Maple would be to allow users to employ LaTeX that gets marked up and placed into a graph. I understand that Matlab already has this feature. In the past I have used Maple to produce graphs, and then used the LaTeX package psfrag to replace text "marker strings" in the eps file with marked up LaTeX. Maple seems to be going hog wild with this document mode thing, which is just horrible and does not work right, and seems oblivious to the power of LaTeX and pretends that it does not exist. Yes, I know about the command latex( ); Indeed it is useful, but it has some fundamental shortcomings and needs rewriting.

A super addition to Maple would be to allow users to employ LaTeX that gets marked up and placed into a graph. I understand that Matlab already has this feature. In the past I have used Maple to produce graphs, and then used the LaTeX package psfrag to replace text "marker strings" in the eps file with marked up LaTeX. Maple seems to be going hog wild with this document mode thing, which is just horrible and does not work right, and seems oblivious to the power of LaTeX and pretends that it does not exist. Yes, I know about the command latex( ); Indeed it is useful, but it has some fundamental shortcomings and needs rewriting.

One very frustrating feature of Maple10 is that "undo" is not at all generous. In Maple 9 and earlier versions, undo was very useful. Now it dies very early. I wonder if it will behave better in Maple 11. [And my pet peeve is that the bread and butter activity of copy/paste stopped working right in Maple 10. It is just maddening. Hope it works better in Maple 11, but I am not optimistic.]

Notice assume(r>0,x1>0,r>x1); f2:=x->sqrt(r^2-x^2)-x1: int(f2(x),x=x1..sqrt(r^2-x1^2)); does work! Maybe by placing global assumptions on r and x1 before defining f2 has something to do with it.

I do see your point, now that I see the integrand was intended to be 1/(2+cos(x)) instead of 1/(1+2*cos(x)) .

The fact that Maple's indefinite integral for 1/(2+cos(x)) is not continuous, at least for real values of x, does kind of stink.

But nevertheless, Maple does seem to give the correct definite integral, so it is not altogether clear to me that the proposed form of the antiderivative, which has the virtue that it is continuous for real values of x, is vastly superior.

Yes, it might be less confusing if we shift the discontinuities of the antiderivative to live off the x-axis, but this might be presumptuous. If one is looking at the family integrate(1/(a+cos(z)),z), then one might actually find virtue to Maple's form of the antiderivative.

Much work is done on R, but much work is also done on C, and poor Maple can't read minds as well as humans can, and so it gets caught in the middle.

> integrate(1/(2+cos(x)),x=2..5);evalf(%);

> F:=x->(x - 2*arctan(sin(x)/(2 + sqrt(3) + cos(x))))/sqrt(3);

F is the proposed nice antideriative, which is continous-at least for real values of x.

> F(5)-F(2);evalf(%);

>

>

The antiderivatives lead to definite integrals that are numerically the same. It is not clear that we need to rock the boat. But it would probably be harmless enough.

Here is one nice way to get your matrix. phi:=j->(x/l)^(j+1); m:=(i,j)->int(phi(i)*phi(j)*p,x=o..l); M:=Matrix(1..5,1..5,m); On the side, use classic Maple10 if you want to be able to copy and paste. The postmodern version of Maple seems to routinely mess up the paste when you copy and paste--the paste gives you some unworkable stuff that looks like MathML or something. I am dumbfouned that Maple has not fixed this. Copy and paste is your basic bread and butter.

Here is one nice way to get your matrix. phi:=j->(x/l)^(j+1); m:=(i,j)->int(phi(i)*phi(j)*p,x=o..l); M:=Matrix(1..5,1..5,m); On the side, use classic Maple10 if you want to be able to copy and paste. The postmodern version of Maple seems to routinely mess up the paste when you copy and paste--the paste gives you some unworkable stuff that looks like MathML or something. I am dumbfouned that Maple has not fixed this. Copy and paste is your basic bread and butter.

My colleague, who works in deformations of algebras, will often give me a system of polynomial equations and ask me to help him find a solution with Maple. He will tell me that he knows by theory that there are solutions. But in fact it seems next to impossible to find them. Once he gave me a system of 3000 polynomials in 8 variables. The system is way over constrained! I was amazed that Maple could not solve the system. So I did go and collect random statistics for systems with these parameters. Typically it does not take Maple long at all to find that there are no solutions. Expected time...not very long. It does not eat up your computer. Try it! But as soon as you give it a comparable system based in research reality, you never get a solution. This is not a flaw in Maple. It is just a statement about mathematics. Interesting problems are really hard. You cannot expect that you can sit back and wait for a computer algebra system to do the insightful work that is required to find a solution when the system of equations is interesting.

My colleague, who works in deformations of algebras, will often give me a system of polynomial equations and ask me to help him find a solution with Maple. He will tell me that he knows by theory that there are solutions. But in fact it seems next to impossible to find them. Once he gave me a system of 3000 polynomials in 8 variables. The system is way over constrained! I was amazed that Maple could not solve the system. So I did go and collect random statistics for systems with these parameters. Typically it does not take Maple long at all to find that there are no solutions. Expected time...not very long. It does not eat up your computer. Try it! But as soon as you give it a comparable system based in research reality, you never get a solution. This is not a flaw in Maple. It is just a statement about mathematics. Interesting problems are really hard. You cannot expect that you can sit back and wait for a computer algebra system to do the insightful work that is required to find a solution when the system of equations is interesting.

For the first example int(1/(1+2*cos(x)),x=2..5); the value >is< undefined if we understand the integral to be the Riemann integral. So Maple's answer is correct. One only attempts to define a Riemann integral for bounded functions on [a,b]. If we want to consider some sort of "Cauchy principal value" then we should first look at int(1/(1+2*cos(x)),x=2..2*Pi/3); which is infinite, as correctly returned by Maple. For the second example: int(abs(exp(x)+cos(x)),x=-2..0); we are just not playing fair if we expect Maple to give a value. The answer is just one of those uncountably many irrational numbers that have not been baptized with a name, other than int(abs(exp(x)+cos(x)),x=-2..0). Popular numbers like e, pi, gamma, sqrt(2) have been baptized. But we only have countably many words in our vocabulary. We could define a number "pu" by saying that pu is equal to int(abs(exp(x)+cos(x)),x=-2..0), point out that it is about 1.846479131, and then train Maple to return "pu" whenever we asked it to evaluate int(abs(exp(x)+cos(x)),x=-2..0) but it would probably not catch on in the community.