Did you have any trouble reverting the substitution? A simple

eval(expr, Sw= Sum(w[k], k= 1..n);

should do it.

If the sum cannot be simplified or be put into closed form, then you should use Sum instead of sum. If you use sum, then Maple wastes time trying to put the sum into closed form everytime the expression is evaluated.

This is an Answer to the amended question in your Reply to your original Question, which is a much more complicated problem than your original Question.

evalindets(B, And(`*`, satisfies(P-> {1/T[1],1/V[2]} subset {op(P)})), P-> P*T[1]*V[2]);

(It looks a lot better in a worksheet than posted here.)

Use the Select Execution Group keyboard shortcut Ctrl-Alt-Shift-E. It's true that that will only select one execution group at a time, but, c'mon, how many execution groups are we talking about moving here? If you have a huge number of execution groups, maybe you should merge them using Join Execution Groups F4.

Note how Maple represents the derivative of tan(x):

diff(tan(x), x);

Based on this, I guessed that your integral might work better if I replaced sec(x)^2 with the above.

int(tan(x)^(n-2)*(1+tan(x)^2), x);

The above answer is returned nearly instantaneously.

I was quite surprised that the following not-quite-simplified answer was also returned nearly instantaneously. I had expected that the integration would be distributed over the sum, leading to an unevaluated answer for powers of tangent.

int(tan(x)^(n-2)+tan(x)^n, x);

Based on the result of my first integral above, I guessed that if I redefined the derivative of tangent, then your integral would evaluate easily. That guess was correct.

eval(`diff/tan`);

restart: #Need to clear diff's remember table.
unprotect(`diff/tan`);
`diff/tan`:= proc(z,t,$) `tools/df_dt`('tan(z)', z-> sec(z)^2, t) end proc:
diff(tan(x), x);

int(tan(x)^(n-2)*sec(x)^2, x);

So this explains why the derivative-divides rule (see Alejandro's Answer) failed to be applied to your original integral.

Maple is interpretting your quoted expressions as mathematical expressions (with division and implied multiplications) because you have used the wrong type of quotation marks. Use either double quotes

"Extracelular urea concentration (mg/dL)"

or back quotes

`Extracelular urea concentration (mg/dL)`

In the future, PLEASE PROVIDE THE COMPLETE CODE needed to generate your solution. I DID NOT ENJOY having to recreate your differential equations so that I could execute your code given above. Maybe that's why it took four days for you to get an Answer.

By the way, cellular is spelled thus, with a double L.

If A is a (1-dimensional) Array, the appropriate subarrays can be extracted by the following intuitive indexing:

A[[1..4,5,8,9]], A[[6,7]], A[[10]];

Note the extra square brackets, which are required for the first two, and are needed to make the last one an Array rather than just a single entry.

In order to map an indexing operation over a list of index specifications, we need an indexing operator in prefix form. Maple provides `?[]` for this purpose. So, `?[]`(A, [Ind]) is the same as A[Ind]. (Note carefully the extra square brackets that are required around the index specifications in the prefix form!)

To add the extra square brackets en masse (by mapping), we need the prefix form of the add-square-brackets operator. That's `[]`. So `[]`~ ([A,B,C]) is the same as [[A],[B],[C]].

The first argument to plot needs to be a list, not an Array, so I use convert(..., list).

Here's an example of the three plots (one color for each group of indices) put together with plots:-display. If you want three separate plots, simply replace plots:-display with print~.

A:= Array(1..10, n-> t-> t^(n-5)):
Inds:= [[1..4,5,8,9], [6,7], [10]]:
plots:-display(
   zip(
        (A,C)-> plot(convert(A, list), -2..2, -32..32, discont, color= C),  
        map2(`?[]`, A, `[]`~(Inds)),
        [red,green,blue]
   )
);

[Edit: It turns out that the first two of the three solution techniques presented in this Answer are irrelevant because the true cause of the problem was the OP's high setting of Digits (which was not revealed until much later in the discussion below). I just leave them here so that the rest of the thread makes sense.]

Here are three completely different techniques to solve this.

 Procedural form of input to Int and plot

Use the procedural form of input to Int and plot, which does not use an independent variable.

Also, your way of extracting the procedure from the dsolve solution is convoluted: There is no need to mention the independent variable. Ordinarily, what you did with the independent variables would make sense to Maple, although it is not necessary to do it. However, the procedures returned by dsolve(..., numeric, output= listprocedure) treat a symbolic independent variable in a slightly strange way: Instead of returning y(x), they return `y(x)`(x). This would be fine if the numeric procedure was named `y(x)`, but you've changed the name to u.

So, to correct the problem, do this:

# No change needed to dsolve.
So:= dsolve(
     [-1.2*diff(y(x), x$2)+0.8*y(x) = 2, y(0)=1, y(1)=0], y(x),
     'numeric', 'output' = listprocedure
):

# Simple extraction; no unapply needed.
u:= rhs(So[2]):

# No independent variables on the procedures or the range.
plot([u, u^2], 0..1);

# No independent variables, and note the capital I
evalf(Int(u, 0..1));

evalf(Int(u^2, 0..1));

 Or... Name the numeric procedure appropriately

You could just use `y(x)` instead of u as the target for the unapply, then make the next line u:= `y(x)`, and keep the rest of your code exactly as it was before:

`y(x)`:= unapply(rhs(So[2])(t), t):
u:= `y(x)`:

 Or... Make the integrals part of dsolve's output

You could simply include the appropriate functions in the original system passed to dsolve. It has often been argued here at MaplePrimes that this is the most numerically robust way to do it, due to dsolve's sophisticated error control. (However, I suspect that it is more computationally intensive because it requires the computation of the integrals at all the intermediary points.) This way involves a little bit of thought to rearrange the integrals into derivative form.

So:= dsolve([
     -1.2*diff(y(x), x$2)+0.8*y(x) = 2, y(0)=1, y(1)=0,

     diff(u(x),x) = y(x), u(0)=0,
     # So u(x) = int(y(t), t= 0..x).

     diff(u2(x),x) = y(x)^2, u2(0)=0
     # So u2(x) = int(y(t)^2, t= 0..x).

     ], [y(x), u(x), u2(x)],
     'numeric', 'output' = listprocedure
):

So(1);

The required integrals are the last two entries in the above list.

Do you mean that you want the display of the function to be J_0(k), i.e., a more standard notation than BesselJ? Then set

interface(typesetting= extended);

Then,

inttrans[fourier](piecewise(abs(x)<1,1/sqrt(1-x^2),0),x,k);

The typesetting level can also be changed from the Tools -> Options -> Display menu.

This is a bug. I'm amazed that it hasn't surfaced before now. Looking at the simple code for IntegrationTools:-Combine shows both the source of the bug and the workaround.

showstat(IntegrationTools:-Combine);

IntegrationTools:-Combine := proc(v)
local ints, x, X, input, output;
   1   ints := indets(v,'And(DefiniteIntegral,Not(MultipleIntegral))');
   2   x := IntegrationTools:-GetVariable(ints[1]);
   3   input := subsindets[':-flat'](v,'And(DefiniteIntegral,Not(MultipleIntegral))',t -> Change(t,GetVariable(t) = X));
   4   output := combine(input,'int');
   5   return subs(X = x,Change(output,X = x))
end proc

So, it is clear from lines 1 and 2 that the command can only work on definite integrals! For indefinite integrals, just use combine(..., int).

combine(Int(sin(x),x)+Int(cos(x),x), int);

plots:-polarplot([r, Pi/6, r=0..2]);

The upper limit for r is arbitrary; I chose 2 for no particular reason. Note that Pi is with a capital P.

This technique is akin to how one would plot a vertical line in rectangular (Cartesian) coordinates:

plot([3, y, y= -2..2]);

I'm not sure if command polarplot is available in Maple 13. If it isn't, I'll come up with something else for you.

Perhaps they meant semicolons, which are used to separate rows:

< 6,8 ; -3,9 ; 4,14 >;

But that's row-major order.

By the way, using the angle-bracket operators uses significantly less processor time than using Matrix or Vector, so I'd recommend using them (or the built-in rtable) in programs whenever possible. Often, it is useful or necessary to use the prefix forms `<,>` and `<|>`. (There is no `<;>`.) For example, to convert a row-major listlist LL into a Matrix, you could do

<`<|>`~(LL)[]>;

But that example is just to illustrate the power of the prefix notation, because instead I'd probably use

rtable(LL, subtype= Matrix);

If your answer to the first question in my Reply above is yes, then this routine will do it:

SeparateLastTerms:= E->
     evalindets(
          E, specfunc(anything,{Sum,sum}),
          S-> eval(op(1,S), op([2,1],S)= op([2,2,2],S)) + subsop([2,2,2]= op([2,2,2],S) - 1, S)
     )
;

Example of use:

E:= Sum(A[k], k= 2..n+1) + sum(f(j), j= 3..m):
SeparateLastTerms(E);

Change the line

evalm(Matrix(n,n,symbol=C))

to

Matrix(n,n, (i,j)-> C[i,j])

The symbol=C option does not work when C has already been assigned values.

There is no need---ever---to use evalm with capital-M Matrices, which are the only type of Matrices that you should be using. Likewise, there is no need for with(linalg).

You wrote:

I understand in Maple one uses the back quote key (or rather the apostrophe, 0X27) to prevent one time evaluation of expression.

The apostrophe is not the same as the back quote! The aposthrope is ASCII character 39 (or 0X27 as you say); the back quote is `, ASCII character 96 (or 0X60). It is important in Maple not to mix these up! They each have very specific uses. The aposthrophe delays evaluation, and the back quote turns arbitrary groups of characters into names.

This might indicate that the front end parser did this simplification before the main evaluator got hold of it, so it was too late?

Yes, that is essentially correct. We refer to it as "automatic simplification." Yes, these cannot be delayed with quotes.

Either way, how would one make Maple return 16/4 when the input is '16/4'?

In Maple 18, use the InertForm package:

InertForm:-Display(InertForm:-Parse("16/4"));

or

InertForm:-Display(`%/`(16,4));

(Note the use of back quotes in the last command.)

You need to choose a branch of the absolute value. Each branch gives a different solution. When there are multiple solutions possible for the highest derivative, numeric dsolve will not choose one of them for you. So, solve the systems y*y''' = -1 and y*y''' = 1 separately and look at the different plots.