It's obvious that the smartview option cannot guess all user's intentions.
If you don't know the relevant y-domain (used by view=...) you may try a slope criterion. E.g.

restart;
f:=exp(-3^(1/2)*(cos(x)-1)/sin(x)):
f1:= diff(f, x):
maxslope:=100:
ff:=piecewise(abs(f1) < maxslope, f, undefined):
plot(ff, x=0..2*Pi);

restart;
Digits := 15: 
nu_0 := sqrt(k^2 - k_0^2): nu_m := sqrt(k^2 - k_m^2): 
pref2 := 2*rho_me*exp(-nu_0*(zs + z))*BesselJ(0, k*sqrt((xs - x)^2 + (ys - y)^2))*k/(rho_me*nu_0 + rho_0*nu_m*tanh(nu_m*d)):

f:= (eval(0.5*(((eval(pref2, [z = 0, k_0 = 0.018371886863098, xs = 0, ys = 0, zs = 10, rho_0 = 1.2, d = 0.04, rho_me = 1.528516816439260 - 1235.297048886680*I, k_m = 0.490806242885258 - 0.490314205803914*I])))), [z = 0, k_0 = 0.018371886863098, xs = 0, ys = 0, zs = 10, rho_0 = 1.2, d = 0.04, rho_me = 1.528516816439260 - 1235.297048886680*I, k_m = 0.490806242885258 - 0.490314205803914*I]) ):

dom := [k=0..100, x = -2/2 .. 2/2, y = -3/2 .. 3/2]:  # 100 is enough
evalf(Int(f, dom, digits=10));

                               0.5867621705 - 0.1097638086*I

The answer is correct, a being a name and non-constant.
eval(diff(y(x,t), x$3), x=Pi);  and  eval(diff(y(x,t), x$3), x=a+1);

will look as you expect.

You may declare a as a constant. In this case the result will appear as for Pi:
constants := constants, a;
eval(diff(y(x,t), x$3), x=a);
            

restart;
N:=100:  dx:=evalf(2*Pi/N):
X:=<seq(k*dx, k=-N/2..N/2)>:
Y:= sin~(X):
f:=unapply(CurveFitting:-Spline(X, Y, x, degree=1),x):
Z:= [seq( evalf(Int(f, X[1]..x, epsilon=1e-5)), x=X)]:
plots:-display(plot(X,Y),  plot(X,Z), color=[red,blue]); 

Use the definition:

P := (x-6)^2 + (y+3)^2 = (x+2*y-1)^2 / 5;

with(plots): display(implicitplot([P, x+2*y-1], x=-10..10,y=-10..10), pointplot([6,-3], color=red));

The singularities of F1 are on the Jordan curve

plots:-implicitplot(1/F1, x=0..k - varepsilon,y=0..2*Pi, view=0..6.5, color=red);

Note that the integrand is even in x, so, one may integrate over x = 0 ... 1 - eps.

Unfortunately Maple cannot compute numerically a CPV integral, but it is easy to transform it into a regular one. If the singular point is c, then 

Int(f(x), x =c-r .. c+r, CPV) = Int(f(c+t) + f(c-t), t = 0 .. r)

Strange implementation (bug?) for numboccur.

restart;
L:=[1., HFloat(1.), 1., 0., 0.];
#                   L := [1., 1., 1., 0., 0.]

numboccur(L, 1.);
#                               2

numboccur(L, 1.0);
#                               0

numboccur(L, HFloat(1.0));
#                               1

evalb(L[1]=L[2]);
#                              true

select(`=`, L, 1.); select(`=`, L, 1.0);
#                          [1., 1., 1.]

#                          [1., 1., 1.]

dismantle(%);

#LIST(2)
#   EXPSEQ(4)
#      FLOAT(3): 1.
#         INTPOS(2): 1
#         INTPOS(2): 0
#      HFLOAT(2): 1.
#      FLOAT(3): 1.
#         INTPOS(2): 1
#         INTPOS(2): 0

arctand ignores the second argument.

arctand(y,x) simplifies to arctand(y)

BTW, for y <> 0,

arctan(y,x) = (Pi/2)*signum(y)-arctan(x/y)

The fact that leadterm is not documented under asympt is probably because asympt(f, x, n) is actually equivalent to series(f, x=infinity, n), and here leadterm appears. By taking n=2, leadterm is not strictly necessary, but it is convenient because O(...) is removed.

There is an old worksheet at the Application Center https://www.maplesoft.com/applications/view.aspx?SID=6322

It is clear enough and includes an animation.

Download extrema.mw

restart;
eq1:= f(x) + 2*f(1/x) + 3*f(x/(x-1)) = x:
eq:=eq1, simplify(eval(eq1, x=1/x)):
eq:=eq, simplify(eval(eq1, x=x/(x-1))):
eq:=eq, simplify(eval(eq1, x=-1/(x-1))):
eq:=eq, simplify(eval(eq1, x=1-x)):
eq:=eq, simplify(eval(eq1, x=(x-1)/x)):
solve([eq], indets([eq],function)): f(x) = eval(f(x), %);
simplify( eval(eq1,  f = unapply(rhs(%), x)) );   # check

So, you want the surface integral on the projection of a cuboid on a plane.

V:=<0,0,0>, <1,2,0>, <0,2,0>, <1,0,0>, <0,0,1>, <1,2,1>, <0,2,1>, <1,0,1>:
T:=<-1,0,0>, <0,-1,0>, <0,0,-1>: 
PM:=LinearAlgebra:-ProjectionMatrix([T[2]-T[1], T[3]-T[1]]):
Pr := v -> T[1]+PM.v:
Pr2:= v -> (PM.v)[[1,2]]:  # ConvexHull works in dimension 2.
P:=Pr~([V]):
P2:=Pr2~([V]):
ind:=ComputationalGeometry:-ConvexHull(Matrix(P2)^+):
C:=P[ind]: n:=nops(C):  # the vertices of the projection
Tri:=(v1,v2,v3) -> Surface( v1 + t*(v2+s*(v3-v2)-v1), s=0..1, t=0..1):
# the parametrization of the interior of a triangle
add(VectorCalculus:-SurfaceInt( x^2+y^2, [x,y,z] = Tri(C[1],C[i],C[i+1])), i=2..n-1);