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INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

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

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INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

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

  2. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

  3. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

  4. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

  5. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Example 3 The Heaviside “Cover-up” Multiply by 1 subsitute 2 Match coeff subsitute

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

  7. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS 1 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 quadratic (irred) q(x)= product of linear factor All distinct Some repeated repeated All distinct case1 case3 case4 case2

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

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

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

  11. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS q(x)= product of quadratic (irred) repeated All distinct case3 case4 Example Expand by partial fraction (DONOT EVALUATE ) Example Expand by partial fraction (DONOT EVALUATE )

  12. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

  13. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS

  14. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS Expand by partial Find the constants Evaluate the integral

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