7. Integration of Rational Functions

Calculus 2



Weiqi Zhou

Rational Funcitons

functions of form ，where are polynomials.

it is called proper if the degree of is less than that of ,
and improper otherwise.

e.g., vs.

Polynomial Divisions

Polynomial Divisions

if are polynomials of order respectively and ， is not identically ，then there is a uniqe pair of polynomials , where the degree of is strictly less than that of ，such that

reduction

only proper rational functions are considered here,
since an improper rational function can be written as a polynomial plus a proper rational function.

Example

  • compute

  • take

  • solve it to get

Example

  • compute



Example

  • compute





The Strategy

decompose into lower and simpler terms

Exercises

  • compute following indefinite integrals





Example

  • compute

  • let ，then ，and



  • ，plug back in

Example

  • compute

  • let ，then ，and



  • ，plug back in

Exercises

  • compute following indefinite integrals





Summary

  • Rational Function and Polynomial Divisions

  • Decompositions

  • Substitutions Involving -th Roots