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

2. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

5. Multiply by Match coeff subsitute 1 2 3 The Heaviside “Cover-up” Example

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

7. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS 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

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

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

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

11. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS q(x)= product of quadratic (irred) All distinct repeated case3case4 Example Expand by partial fraction (DONOT EVALUATE )

12. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

14. Expand by partial Find the constants Evaluate the integral