1. MAT 1235 Calculus II Section 7.4 Partial Fractions http://myhome.spu.edu/lauw

2. HW Please do your HW ASAP. Please actually do your HW.

3. Partial Fractions Review PF from pre-calculus Use PF to simplify integrands Break up a complicated rational function into smaller ones Each of the smaller rational function is easier to integrate

4. Preview Use Partial Fractions to decompose a rational function into a sum of simpler rational functions

5. Assumption We assume: deg(P(x))<deg(Q(x)) If this is not the case, we can use long division to rewrite the rational function as

6. Assumption We assume: deg(P(x))<deg(Q(x)) If this is not the case, we can use long division to rewrite the rational function as

7. Example 1

8. Remark In stead of using the substitution, we can use the following formula

9. Example 2

10. Assumption We assume: deg(P(x))<deg(Q(x)) Depends on the form of Q(x), we have 3 different cases.

11. Case I Q(x) is a product of distinct linear factors

12. Case I Q(x) is a product of distinct linear factors

13. Example 3

14. Expectation Make sure you write down the final partial fractions (on the right hand side) before you proceed to evaluate the integral.

15. To save time… We will work on the partial fractions only We are not going to actually complete the integration

16. Case II If (ax i +b i ) is repeated r times

17. Case II If (ax i +b i ) is repeated r times, we use

18. Example 4

22. Case III If Q(x) has a irreducible factor ax 2 +bx+c,

23. Case III If Q(x) has a irreducible factor ax 2 +bx+c, we use

24. Example 5