put the plot in a variable, e.g., p1:=plot(x^2,x=0..3):

and then lprint(p1);

Even though the first two seem to be interpreted as 2-D integrals, the correct syntax for 2D integrals needs the variables and ranges in a list (or nested 1D integrals), like

simplify(Int(integrand, [phi = 0 .. 2*Pi, r = 0 .. 2]));

and then you get the correct answers. 

You have two first order equations, so if you know omega[0] and sigma, you need only two initial conditions. Put them as a sequence not as a list to solve the syntax error. So give omega[0] and sigma values then

ICS:=C(0)=0,A(0)=1:
sol:=dsolve({C_t,A_t,ICS},{C(t),A(t)},type=numeric):
pRange:=0..20:
plots:-odeplot( sol, [t, A(t)],t=pRange, numpoints=10000 );  #note A(t) not A

If you want to find the values of omega[0] and sigma by specifying two additional conditions, then I think that only works with boundary value problems, and you have only initial conditions. Note also the syntax for derivative conditions

is D(C)(0)=0 for derivative zero at time zero 

I think dchange existed in Maple V, so try

integral := Int(2*(sin(theta)/cos(theta))^(2*p-1), theta = 0 .. (1/2)*Pi);
PDEtools[dchange]({theta=arctan(t)},integral);
value(%);

gives Pi*csc(Pi*p) (in Maple 2017)

add the option range=-1..1 as the second argument to HeatMap works for me in Maple 2017.

The 'ps' driver (not 'eps) is supposed to be uncolored by default, but didn't seem to be. Using color=none is black and white,  but for me it produces a black background with white text and lines. Using color=gray gave a gray plot line, but no axes or text.

So adding color="Black" in the display command to override default plot colors should be an alternative, but if colors were given in the plot command, then you probably have to edit the plot structure.
 

The general solution can be (partly) obtained as below. Here sqrt(_c[1]) is n, but to get this you need to know that the function is periodic in theta (trying u(r,theta)=u(r,theta+2*n*Pi) doesn't work). To keep the solution finite at the centre of the disk you have to set _C2=0, and again you have to manipulate with Maple further to decide this.

Download PDE2.mw

Using HINT=`*` forces the product form, and then you can implement another BC. Edit: added the 2nd BC "by hand"

Download PDE.mw

As the others have implied, a lot depends on what you consider simple. For the last one I like

Student[Precalculus]:-CompleteSquare(k^2+2*k+9); which gives (k+1)^2+8

and for the rational functions I like continued fractions:

convert((k-2)*(k^2+5)*(k^3-k^2+7*k+8)/(6*k*(k^2-3*k+8)),confrac,k);

The second sqrt argument k^4-10*k^3+37*k^2-60*k+180 can be simplified to (k-2)^2*(k-3)^2+144. The first one is harder.

Download sqrts.mw

Your answer to @Preben Alsholm  suggests you are using a table, not an array. Then the following works:

Download Occurrences.mw

Under help for list (in Maple 2017.3), we find:
"Lists and sets can be nested, in which case selection can be done in one of two ways:  S[i][j]...[n] or S[i,j,...,n]."

This seems to allow the interpretation of m[[1,2],2] as m[[1,2]][2] though it is very confusing.

The use of f(x) and df(x) is confusing in this context. And the print is redundant. Just use:

restart;
f:=x^2-3;
df:=diff(f,x);
x:=1.;  #use floating point here
for i from 2 to 5 by 1 do
    x:=x-(f/df);
    end do;

Your second approach works if you use solve:

Download linear.mw

If I paste your code into an execution group and hit enter, I see the cursor just before R on the line starting: locals R (It's quite hard to see). This will be near the error, which is that "locals" should be local. Continuing and fixing some missing semicolons gives a syntax error free code:

 # ctrl + del to delete a Maple cell
 # golden search implementation
 # chapra 7th ed
 golden_search := proc(f, xl, xu, es100, maxiter)
 description "find the optimal point using golden-search optimization method";
 local R, d, xopt, x1, x2, f1, f2, iter, ea, xint;
 R := (sqrt(5)-1)/2;
 d := R*(xu-xl);
 x1 := xl + d;
 x2 := xu - d;
 f1 := evalf(eval(f, x = x1));
 f2 := evalf(eval(f, x = x2));
 # declare iterator and ea
 iter := 1;
 ea := 1.00;
 # start while
 while ea*100 > es100 and iter < maxiter do
     d := R*d; # new golden ratio
     xint := xu - xl;
     if f1 > f2 then
         xopt := x1;
         fx := f1;
         x1 := x2;
         x2 := x1;
         x1 := x1 + d;
         f2 := f1;
         f1 := evalf(eval(f, x = x1));
     else
         xopt := x2;
         fx := f2;
         xu := x1;
         x1 := x2;
         x2 := xu - d;
         f1 := f2;
         f2 := evalf(eval(f, x = x2));
     end if;
     # calculate new ea
     if xopt <> 0 then
         ea := (1 - R)* abs(xint/xopt);
     end if;
     iter = iter + 1;
 end do;
 # return xopt and fx here
 xopt;
 end proc: