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

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

3. Use long division 1 Factor q(x) as linear factors or irreducible quadratic 3 Express p(x)/q(x) as a sum of partial fraction 4 q(x)= product of linear factor All distinctSome repeated q(x)= product of quadratic (irred) All distinct repeated case1 case2 case3case4 Check if we can use subsitution 2

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

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

6. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS q(x)= product of quadratic (irred) All distinct repeated case3case4 Example Find

7. Sec 7.4: INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS q(x)= product of quadratic (irred) All distinct repeated case3case4 Example Express only Example Find

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

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