Dear student,
The eccentricity of an  ellipse is defined as the ratio of the distance between any point in the  ellipse and the focus and distance between point and a fixed line  called directrix.
 
            e = distance between P and F/distance between P and line D (Directrix)
Formula for eccentricity :      
               Formula for calculating e is
                b2 = (a2)(e2-1)    So e2- 1 = b2/a2     or e2 = b2/a2 +1 
                e = √(b2/a2+1)
                Eccenticity = square root of square of semi minor axis divided by squares of semi major axis
						

							
eccentricity of hyperbola = 1+b2/a2 = e         ..............1
eccentricity of conjugate = 1+a2/b2 = ec2        ..................2    (ec = eccentricity of conjugate hyperbola)
 
from these two equatns , ec  = e/(e2-1)1/2            ......................3
f(f(e)) = e                          (n = 2 even)
f(f(f(e))) = e/(e2-1)1/2       (n = 3 odd)
 
now , if n = even then
integral ede = e2/2 lim from 1 to 3
                  =4                                   (A)
if n = odd then
 integral  ede/(e2-1)1/2  lim 1 to 3
      = (e2-1)1/2          lim 1 to 3
       =2root2                                     (D)
 
hence option A,D are correct ....
 
approve if u like my ans