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

INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

Multiply by Match coeff subsitute The Heaviside “Cover-up” Example

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

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

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

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:

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

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

INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

Expand by partial Find the constants Evaluate the integral