The required integral has the form:
First, consider a cubic polynomial of the form:
then according to the methodology detailed in [ 42 ] define the following variables :
Now, three cases must be considered :
y 2 N ˆ’ h 2 > 0;
y 2 N ˆ’ h 2 = 0;
y 2 N ˆ’ h 2 < 0.
y 2 N ˆ’ h 2 > 0 . Here, the cubic polynomial has only one real root, given by:
Hence:
where ¼ = + x 1 and = .
Applying this to (A.2.) the integral takes the form:
where:
Further:
where:
and so:
where · = ½ ¼ and = .
y 2 N ˆ’ h 2 = 0 . For this case there are three real roots, with a repeated root. Again from the methodology in [ 42 ], the three roots are:
Substituting into (A.2.) we have:
where:
y 2 N ˆ’ h 2 < 0 . Here we have three distinct real roots. These roots are:
where [7] :
Again, substituting into (A.2.), we evaluate the integral as:
where:
and
[7] As defined in (A.3.d), h = 2 d 3 .