A[r_,theta_]:=(
a=r*Sin[(Pi-theta)/2]/Sin[theta/2];
b=r*Sin[theta/2]/Sin[(Pi-theta)/2];
(a+b)*r/2
)
A[r,2*Pi/k]

B[l_,r_,theta_]:=(
s=r*Sin[Pi/2-theta]/Sin[3*theta/2];
t=l*Sin[(Pi-theta)/2]/Sin[3*theta/2];
q=t*Sin[theta]/Sin[theta/2]-s;
h=q*Sin[Pi/2-theta];
y=q*Sin[3*theta/2]/Sin[(Pi-theta)/2];
(1/2)*y*h
)
B[l,r,2*Pi/k]

f[l_,theta_]:=(
Integrate[(1/l)*E^(-A[r,theta]-B[l,r,theta]), {r,0,l}]
)
f[l,2*Pi/k]

Q[x_,theta_] := 1-Exp[-x^2*Sin[(Pi-theta)/2]/Sin[theta/2]]
Q[x,theta]

p:=Function[{x,theta},D[Q[x,theta],x]]
p[x,alpha]

Integrate[p[l,2*Pi/k]*f[l,2*Pi/k],{l,0,Infinity}]

Plot[Out[-1],{k,5,500}]

Limit[Out[-2],k->Infinity]

N[Out[-1],10]

N[Out[-4]/.k->{5,7,9,11,13,15,17,19},10]

N[k*(2-Out[-5])/.k->{5,7,9,11,13,15,17,19},10]