Stewart Calculus Section 7.4

1. 7.4 Integration of
Rational Functions
Rational function: ratio of polynomials.
Ex: + +
+

2. 7.4 Integration of
Rational Functions
Rational function: ratio of polynomials.
Ex: + +
+
Fact: A rational function can always be
decomposed to summation of terms like
+
polynomials or ( + ) or ( + + ) where
A, B, etc. are constants.
Ex: − + +
= + + + −
− − + − ( − ) +
Partial Fractions

3. First step: if the rational function is
improper (the degree of numerator is more
than or equal to the degree of dominator),
use long division to decompose it to a
polynomial plus a proper rational function.
+
Ex: = + + +

4. First step: if the rational function is
improper (the degree of numerator is more
than or equal to the degree of dominator),
use long division to decompose it to a
polynomial plus a proper rational function.
+
Ex: = + + +
As a result:
+
= + + + | |+

5. Second step: factor the dominator of the
proper rational function to be a product of
linear factors and/or irreducible quadratic
factors.
+ + is irreducible if < .
Ex: + = ( + )
+ = ( )( + )
+ + =( ) ( + )

6. Third step: decompose the proper rational
function to be a sum of partial fractions by
method of undetermined coefficients.
+
Ex:
+

7. Third step: decompose the proper rational
function to be a sum of partial fractions by
method of undetermined coefficients.
+
Ex:
+
We factor + = ( )( + )

8. Third step: decompose the proper rational
function to be a sum of partial fractions by
method of undetermined coefficients.
+
Ex:
+
We factor + = ( )( + )
+
so assume = + +
+ +

9. Third step: decompose the proper rational
function to be a sum of partial fractions by
method of undetermined coefficients.
+
Ex:
+
We factor + = ( )( + )
+
so assume = + +
+ +
( + + ) +( + )
=
+

10. Third step: decompose the proper rational
function to be a sum of partial fractions by
method of undetermined coefficients.
+
Ex:
+
We factor + = ( )( + )
+
so assume = + +
+ +
( + + ) +( + )
=
+
thus = , = , =

11. CASE I: the dominator is a product of
distinct linear factors.
CASE II: the dominator is a product of
linear factors, some of which are repeated.
CASE III: the dominator contains distinct
irreducible quadratic factors.
Case IV: the dominator contains repeated
irreducible quadratic factors.

12. +
Ex: find
+

13. +
Ex: find
+
We already knew that
+
= +
+ ( ) ( + )
C AS E I

14. +
Ex: find
+
We already knew that
+
= +
+ ( ) ( + )
+ C AS E I
so
+
= | |+ | | | + |+

15. Ex: find
+

16. Ex: find
+
We factor that + =( ) ( + )
CAS E II

17. Ex: find
+
We factor that + =( ) ( + )
CAS E II
so we assume
= + +
+ ( ) +

18. Ex: find
+
We factor that + =( ) ( + )
CAS E II
so we assume
= + +
+ ( ) +
and find = , = , =

19. Ex: find
+
We factor that + =( ) ( + )
CAS E II
so we assume
= + +
+ ( ) +
and find = , = , =
thus
+
= | | | + |+