Stewart Calculus Section 7.4

1. 7.4 Integration of
Rational Functions
First step: if the rational function is
improper, decompose it to a polynomial plus
a proper rational function.
Second step: factor the dominator to be a
product of linear factors and/or irreducible
quadratic factors.
Third step: decompose the proper rational
function to be a sum of partial fractions.

2. 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.

3. Ex: find
+

4. Ex: find
+
I
CA S E II
We recognize + is irreducible.

5. Ex: find
+
I
CA S E II
We recognize + is irreducible.
Complete the square: + =( ) +

6. Ex: find
+
I
CA S E II
We recognize + is irreducible.
Complete the square: + =( ) +
= ( + )
− =
+ +

7. Ex: find
+
I
CA S E II
We recognize + is irreducible.
Complete the square: + =( ) +
= ( + )
− =
+ +
=
+ +

8. Ex: find
+
I
CA S E II
We recognize + is irreducible.
Complete the square: + =( ) +
= ( + )
− =
+ +
=
+ +
= ( + ) +

9. Ex: find
+
I
CA S E II
We recognize + is irreducible.
Complete the square: + =( ) +
= ( + )
− =
+ +
=
+ +
= ( + ) +
= ( + ) +

10. +
Ex: find
+

11. +
Ex: find
+
+ +
We assume = + CAS E III
+ +

12. +
Ex: find
+
+ +
We assume = + CAS E III
+ +
and solve = , = , = .

13. +
Ex: find
+
+ +
We assume = + CAS E III
+ +
and solve = , = , = .
+
so = +
+ + +

14. +
Ex: find
+
+ +
We assume = + CAS E III
+ +
and solve = , = , = .
+
so = +
+ + +
= | |+ ( + ) +

15. + +
Ex: find
( + )

16. + +
Ex: find
( + ) CAS E
IV
+ + + +
We assume = +
( + ) + ( + )

17. + +
Ex: find
( + ) CAS E
IV
+ + + +
We assume = +
( + ) + ( + )
and solve = , = , = , = .

18. + +
Ex: find
( + ) CAS E
IV
+ + + +
We assume = +
( + ) + ( + )
and solve = , = , = , = .
+ + +
so =
( + ) + ( + )

19. + +
Ex: find
( + ) CAS E
IV
+ + + +
We assume = +
( + ) + ( + )
and solve = , = , = , = .
+ + +
so =
( + ) + ( + )
= ( + )+ + +
( + )

20. +
Ex: find

21. +
Ex: find
√
Let = +

22. +
Ex: find
√
Let = +
zation
Rati onali
+
then =

23. +
Ex: find
√
Let = +
zation
Rati onali
+
then =
= +

24. +
Ex: find
√
Let = +
zation
Rati onali
+
then =
= +
= + | | | + |+

25. +
Ex: find
√
Let = +
zation
Rati onali
+
then =
= +
= + | | | + |+
+
= + + +
+ +