To simplify the equations we will define integral 1
I(m)=
ó õ
c
b
Ö
(d-t)(c-t)(t-b)(t-a)
dt
t
.
(4)
Using a similar notation we will define:
I(e0)=
ó õ
c
b
dt
Ö
(d-t)(c-t)(t-b)(t-a)
,
(5)
I(3 e1)=
ó õ
c
b
(t-a)5/2dt
Ö
(d-t)(c-t)(t-b)
,
(6)
I(2 e1)=
ó õ
c
b
(t-a)3/2dt
Ö
(d-t)(c-t)(t-b)
,
(7)
I(e1)=
ó õ
c
b
Ö
t-a
dt
Ö
(d-t)(c-t)(t-b)
,
(8)
I(-e1)=
ó õ
c
b
dt
Ö
(d-t)(c-t)(t-b)
(t-a)3/2
,
(9)
I(-e5)=
ó õ
c
b
dt
Ö
(d-t)(c-t)(t-b)(t-a)
t
.
(10)
A partial fraction expansion of the integral 4 gives this result:
I(m)=
I(3 e1)-(d+c+b-2 a) I(2 e1)+(d (c+b-a)
+(c-a) (b-a)) I(e1)+dcbaI(-e5)-dcbI(e0)
(11)
The integral I(3 e1) and I(2 e1) are not basic integrals listed
in my table of integrals. They need to be expanded in terms of basic integrals.
First expand I(2 e1).
I(2 e1)=
(d+c+b-3 a) I(e1)-(d-a) (c-a) (b-a) I(-e1)
2
.
(12)
Expand I(3 e1).
I(3 e1)=
3
4
(d+c+b-3 a) I(2 e1)-
1
2
(d (c+b-2 a)+c (b-2 a)
-
1
4
a (2 b-3 a)) I(e1)+
1
4
(d-a) (c-a) (b-a) I(e0)
(13)
Substitute for I(2 e1) in equation 13 using equation 12.
I(3 e1)=
-
3
8
(d+c+b-3 a) (d-a) (c-a) (b-a) I(-e1)+
1
8
(3 d2
+2 d (c+b-5 a)+3 c2+2 c (b-5 a)+3 b2-10 ba+15 a2) I(e1)
+
1
4
(d-a) (c-a) (b-a) I(e0)
(14)
Substitute for I(2 e1) and I(3 e1) in equation 11
using equation 12 and equation 14.