1. Find Find INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Example
Find
Example
Find

2. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

3. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

4. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

5. 3 1 2 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example
Multiply by 3
The Heaviside
“Cover-up” 1
subsitute
Match coeff 2
subsitute

6. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
linear factor
irreducible
reducible
quadratic
factor
quadratic
factor
all factors are irreducible

7. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS1
Use long division 2
Check if we can use subsitution 3
Factor q(x) as linear factors or irreducible quadratic
Express p(x)/q(x) as a sum of partial fraction 4
q(x)= product of linear factor
q(x)= product of quadratic (irred)
All distinct
Some repeated
repeated
All distinct
case1
case3
case4
case2

8. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
Example
Find
All distinct
Some repeated
case1
case2
Example
Cover-up A B C

9. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

10. case1 case2 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
All distinct
Some repeated
case1
case2
Example
Example
Partial
Fraction:
Long
Division:
Factor:

11. case1 case2 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
All distinct
Some repeated
case1
case2
Example

12. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

13. case3 case4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of quadratic (irred)
repeated
All distinct
case3
case4
Example
Expand by partial fraction (DONOT EVALUATE )
Example
Expand by partial fraction (DONOT EVALUATE )

14. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
081

15. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
952

16. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

17. case1 case2 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
Example
All distinct
Some repeated
Long
Division:
case1
case2
Factor:
Remark: only use subsitue method or match coeff to find the constants A,B,C,D,E.
Partial
Fraction:
Multiply:
subsitute:
Integrate:

18. F141 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Multiply by (denominator)
Find the constants
Sum of the coeff
F141

19. F142 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Expand by partial
F142
Find the constants
Evaluate the integral
Evaluate the integral
substitution
tan^-1

20. Find INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
RATIONALIZING SUBSTITUTIONS
Some nonrational functions can be changed into rational functions
Example
Find

21. rational function of sin and cos
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
rational function of sin and cos
952 Dr.Aldaffa 1 2
Use the following to convert integral into rational function in t
Evaluate the integral then back to x’s by:

22. rational function of sin and cos
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
rational function of sin and cos 1 2
Use the following to convert integral into rational function in t
Evaluate the integral then back to x’s by:

24. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Remark: make sure that
1) Deg( numerator ) < Deg( denominator )
Each factor is irreducible
No cancelations ( num & denom )
How to figure that a quadratic factor is irreducible
Discriminant is negative

25. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

26. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

27. case1 case2 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
Example
All distinct
Some repeated
Long
Division:
case1
case2
Factor:
Remark: only use subsitue method or match coeff to find the constants A,B,C,D,E.
Partial
Fraction:
Multiply:
subsitute:
Integrate:

28. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Expand by partial
Find the constants
Evaluate the integral
subsitution
tan^(-1)

29. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS