In your picture you have finite sums. In this case the formula is OK, provided that the rhs exists (i.e. each function is integrable).

For infinite sums (N = oo) the situation is much more complicated. In general, Maple may apply the formal formula but it is the user's responsibility to check whether the result is correct (there are standard theorems covering this, e.g. the dominated convergence theorem). Note also that the validity of the formula depends on the integral being used (Riemann, generalized Riemann, Lebesgue, ...).

int(1/x, x = 1 .. t, AllSolutions);

restart;
M:=2; k:=1;
U:=Matrix(M+1, (r,s) -> `if`((r>1) and (s<r) and (r+s)::odd, 2^(k+1)*sqrt((2*r-1)*(2*s-1)), 0));
DD:=LinearAlgebra:-DiagonalMatrix([U $ 2^k]);

To compute the limit of a Riemann sum, the command eulermac usually works.

But your sum Sum(csc(Pi*x/i)^3*sin(Pi*x)^3/i^3, i = 1 .. n) is not a Riemann sum. And its limit for n-->infinity is actually the series

Sum(csc(Pi*x/i)^3*sin(Pi*x)^3/i^3, i = 1 .. infinity);

It's easy to see that this series diverges for each x in R \ Z (for x in Z the sum is not defined).

So, you have to rethink your problem.

The error message is clear: you must have numeric values for y, because Analytic is designed for expressions of a single variable.
Try for example:
RootFinding:-Analytic(eval(F, y=3), z, re = -5 .. 0, im = -100 .. 100);

The main difference between

f := x -> expr1;
and
g := unapply(expr2, x);

is that expr2 is evaluated, but expr1 is not.

So, for example

restart;
y:=x+1;
f:=x -> y;
g:=unapply(y,x);
y:=7;
f(10);  # 7
g(10);  # 11

For the  x[...], part,  a procedure cannot have indexed names as formal parameters. That is why, unapply automatically corrects e.g.  x[1] into a x_1.  Trying to define  f := proc(x[1]) ... end proc  generates an error.

UMG:=proc(n::posint); # Unitary Matrix Generator
local RM:='LinearAlgebra:-RandomMatrix(n, generator=rand(0. .. 1.))', S, i;
S:=eval(RM+I*RM);
Matrix(LinearAlgebra:-GramSchmidt([seq(S[..,i],i=1..n)],normalized));
end:

Digits:=15:
U:=UMG(3):
fnormal(U.U^*);  #check

restart;
with(Statistics):
p:= 9/10:
L := Distribution(ProbabilityFunction = (k -> -1/ln(1-p)*p^k/k), 'Support' = 1 .. infinity):
X := RandomVariable(L):
S:=Sample(X, 10000):
Histogram(S);

You need to know to write mathematical expressions in Maple:

(1 + a * cos(theta1*x)) * cos(theta2*x) = b;

I'll assume that theta1, theta2 are not 0 (the fact that theta2>theta1 is irrelevant).

If a=0 or b=0 then Maple can solve it easily. Try e.g.

solve( (1+a*cos(theta1*x))*cos(theta2*x) = 0, x, allsolutions );

Now, if a*b <> 0, the equation can be solved symbolically only if  theta1/theta2 is rational (it can be reduced to a polynomial equation and Maple will be able to solve it). Try:

solve( (1+a*cos(theta*x))*cos(2*theta*x) = b, x, allsolutions );

The expensive ifactors should be avoided.

P11:=proc(N)
# divisible by a prime > 11
local n:=N, q;
while irem(n,2,'q')=0 do n:=q od:
while irem(n,3,'q')=0 do n:=q od:
while irem(n,5,'q')=0 do n:=q od:
while irem(n,7,'q')=0 do n:=q od:
while irem(n,11,'q')=0 do n:=q od:
evalb(n>11)
end:

Check11:=proc(a,b);
local n;
for n from a to b do
  if P11(n) then next 
  elif P11(n+1) then n:=n+1; next
  elif P11(n+2) then n:=n+2; next
  elif P11(n+3) then n:=n+3; next fi;
  return n;
od;
true
end:

CodeTools:-Usage(Check11(1000000,2000000));

The help file mentions only  D(y)(x0) = y0 for initial conditions in dsolve.

I see  eval(diff(y(t),t),t=x0)=y0 just as a bonus. So, not a bug!

Just replace Norm(foo, 2)  with  Norm(foo, 2, conjugate=false)

It's not a bug.
expr mod p   ==>  mod is applied to the rational coefficients of the expression; expr is usually a polynomial or a rational expression (i.e. ratio of two polynomials).
So, 
10*i mod 7;   #  ==> 3*i
i mod 3;        # ==> i
12/13 mod 10;   # ==> 4,  because 4*13 = 12  (modulo 10)

with(plots): with(plottools):
display(sphere([0, 0, 0], 1), 
        plot3d(1.01, theta=0..Pi/2, phi=0..Pi/2, coords=spherical, color=red, style=surface),
        axes = framed );