Maple does evaluate the integrals: It says that the answer is undefined, and that is the correct answer. (In your post, you incorrectly use eval on the integrals. But you have the correct evalf in your attached worksheet.) The plots in your worksheet clearly show that the limits are oscillatory, hence undefined.

If Maple could not the evaluate the integrals, then it would've returned the integrals unevaluated.

See ?combinat,randcomb .

Enter the sum like this

Sum((k^2+k-1)/(k+2)!, k= 1..infinity);
value(%);

You should get the answer 1/2.

I don't understand your integral. Clearly it diverges if taken literally:

assume(A>0);
Int(X/(X-A)*(X-1), X= 0..infinity) ;
or
Int(X/(X-A)/(X-1), X= 0..infinity) ;

Either way it diverges. So, could you write it more clearly?

This is how to step through the basis, setting each basic variable to 1 and the others to 0, and evaluating your Zs in light of this. If you post an actual example, I'll show you how to apply this to it.

Sol:= solve(sys, vars):
Basic,NonBasic:= selectremove(evalb, Sol):
Basic:= op~(Basic):
for v in Basic do
     Evals:= {v= 1} union (Basic minus {v} =~ 0);
     eval(Zs, eval(NonBasic, Evals) union Evals)
od:

Eq:= (x^2+y^2)^2=(x^2-y^2):
subs(x= r*cos(theta), y= r*sin(theta), Eq);

combine(%);

%/r^2;

Change the return line to

return convert(orbit, list)

The result needs to be a string (or a name), because an integer can't have the leading zeros. Here's how to make the string:

b:= cat("", op~(convert(a, listlist))[]);

TypeTools:-AddType(Point, [realcons,realcons]):
~set:= x-> convert(x, set):
~list:= x-> convert(x, list):

ThreeIneqs:= proc(A::Point, B::Point, C::Point, XY::[name,name])
uses G= geometry;
local p, a, b, c, L, S:= {A,B,C};
     seq(
          (Eq-> `if`(eval(Eq, XY=~p) < 0, 1, -1)*Eq <= 0)
          (lhs(G:-Equation(
               G:-line(L, (P-> [G:-point(a, P[1][]), G:-point(b, P[2][])])(S minus {p})),
               XY
          ))),
          p= S
     )
end proc:

NumbTrue:= proc(S::~list(boolean), Evals::{set,list}(name=anything))
local r;
     r:= is~(eval(S,Evals));
     `if`(has(r, FAIL), FAIL, numboccur(r, true))
end proc:

TwoOrMore:= proc(Tri::~set(Point), P::Point)
local x,y;
     evalb(NumbTrue([ThreeIneqs(Tri[], [x,y])], [x,y]=~P) > 1)
end proc:

Example:
   
TwoOrMore([[2,3], [4,5], [8,2]], [4,4]);

                                          true

You have two errors:

1. Pi must be spelled with capital P and lowercase i,
2. polarplot should be plots:-polarplot.

It can be done with geom3d:-intersection. In particular, if the result of this command is not NULL, then the input objects intersect.

It can be done in two steps:

convert(s, trig);
combine(%);

f:= 3+2*x+y+2*x^2+2*x*y+3*y^2 = 0:
d1:= implicitdiff(f, y, x);

d2:= implicitdiff(f, y, x$2);

This can be simplified substantially using the original expression and simplify with side relations:

simplify(d2, {f});

diff(f(x,y)=exp(2*x+cos(y)), y$3, x$5);

Edit menu => Split or Join => Split Execution Group (F3)

Simply apply command factor to the result returned by intsolve:

intEq:= 2*cosh(x)-sinh(x)-(2-x)= 1 + int( (2-x+t)* phi(t),t=0..x):
Sol:= intsolve(intEq,phi(x)):
factor(Sol);