
Integration of Algebraic Rational Functions 13-11
Consider,
x x
x x x
2
4 33
1 2 3
+ −
+ − −( )( )( )
( )Step 4
The above rational function is a proper rational function. Integration is same as Case 1.
Assign (x + 1), (x - 2), (x - 3) to partial fractions
A
x +1
,
B
x -2
and
C
x -3
.
Consider the equation
x x
x x x
A
x
B
x
C
x
2
4 33
1 2 3 1 2 3
+ −
+ − −
=
+
+
−
+
−( )( )( )
(1)
=
− − + + − + + −
+ − −
=
−
A x x B x x C x x
x x x
A x
( )( ) ( )( ) ( )( )
( )( )( )
(
2 3 1 3 1 2
1 2 3
5
2
xx B x x C x x
x x x
+ + − − + − −
+ − −
6 2 3 2
1 2 3
2 2
) ( ) ( )
( )( )( )
Equating the numerators on both sides, we get
x x A x x B x x C x x
2 2 2 2
4 33 5 6 2 3 2+ − = − + + − − + − − ( ) ( ) ( ).
Comparing ...