1. Find Find Find Find Find
Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
Rational function:
Example
Example
Find
Find
Example
Find
Example
Example
Find
Find

2. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

3. Sec 7.4: 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
case2
case3
case4

4. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
Example
Find
All distinct
Some repeated
case1
case2
Example
Find

5. Find Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of linear factor
All distinct
Some repeated
case1
case2
Example
Example
Find

6. Find Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of quadratic (irred)
repeated
All distinct
case3
case4
Example
Example
Find

7. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
q(x)= product of quadratic (irred)
repeated
All distinct
case3
case4
Example
Example
Express only
Example
Find

8. rational function of sin and cos
Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
rational function of sin and cos 1
Use the following to convert it into rational function 2
Evaluate the integral then use:

9. Find Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
RATIONALIZING SUBSTITUTIONS
Some nonrational functions can be changed into rational functions
Example
Find

10. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

11. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

12. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS