PHY 480 - Computational Physics
Michigan State University, Spring Semester 2006
INTEGRALS I
Basic formulae
ó
õ
x
°
+ h
x
°
f(x) d x = h f(x
°
+ h)
-
h
2
2
f
¢
(
x
) + ... ,
x
Î
(x
°
,x
°
+h)
(1)
ó
õ
x
°
+ h
x
°
f(x) d x =
h
2
( f(x
°
) +f(x
°
+ h) )
-
h
3
12
f
¢¢
(
x
) + ...
(2)
ó
õ
x
°
+ 2 h
x
°
f(x) d x =
h
3
( f(x
°
) +4 f(x
°
+ h) + f(x
°
+ 2 h) )
-
h
5
90
f
(iv)
(
x
) + ...
(3)
ó
õ
x
°
+ 3 h
x
°
f(x) d x =
3 h
8
( f(x
°
) +3 f(x
°
+ h) + 3 f(x
°
+ 2 h) + f(x
°
+ 3 h) )
-
3 h
5
80
f
(iv)
(
x
) + ...
(4)
ó
õ
x
°
+ 4 h
x
°
f(x) d x
=
2 h
45
( 7 f(x
°
) +32 f(x
°
+ h) + 12 f(x
°
+ 2 h) + 32 f(x
°
+ 3 h) +7 f(x
°
+ 4 h) )
-
8 h
7
945
f
(vi)
(
x
) + ... (5)
Formulae with derivatives
ó
õ
x
°
+ h
x
°
f(x) d x =
h
2
( f(x
°
) +f(x
°
+ h) ) +
h
2
12
( f
¢
(x
°
)
-
f
¢
(x
°
+ h) ) +
h
5
720
f
(iv)
(
x
) + ...
(6)
ó
õ
x
°
+ 2 h
x
°
f(x) d x =
h
15
( 7 f(x
°
) +16f(x
°
+ h) + 7 f(x
°
+ 2 h) ) +
h
2
15
( f
¢
(x
°
)
-
f
¢
(x
°
+ 2 h) ) +
h
7
4725
f
(vi)
(
x
) + ...
(7)