Integration Using Partial Fractions (College Board AP® Calculus BC)

Integration using partial fractions

What are partial fractions?

• A rational function can be written as the sum of partial fractions provided:

  • and are polynomials

  • the degree of is less than the degree of

• Each partial fraction has the following properties:

  • the denominator is a factor of

  • the degree of the denominator will be less than the degree of

  • the degree of the numerator will be less than the degree of the denominator

• In this course will be a product of distinct linear factors

  • Usually only two factors

  • The numerators of the partial fractions will be constant

• For example,

How can I write a rational function as a sum of partial fractions?

• STEP 1
  Factorize the denominator

  • e.g.

• STEP 2
  Write as a sum of partial fractions

  • The numerators are unknown constants

  • The denominators are the linear factors

    • e.g.

• STEP 3
  Multiply both sides by the denominator of the original fraction

  • This gets rid of all the fractions

    • e.g.

    • which simplifies to

• STEP 4
  Find the values of the unknown constants

  • One method is to substitute the roots of the denominators into the equation

    • e.g. Substitute

      

    • e.g. Substitute

      

  • An alternative method is to compare the coefficients of the equation

    • e.g. Collect like-terms on the right-hand side
      

    • Form two simultaneous equations

      

    • Solve to get and

• STEP 5
  Write out the partial fractions

  • e.g.

Can I use partial fractions if the degree of the numerator is not smaller than the degree of the denominator?

• You can use long division to write a rational function as the sum of a polynomial and another rational function

  • e.g.

• You can then write the new rational function as a sum of partial fractions

  • e.g.

How do I integrate using partial fractions?

• It is straightforward to integrate a rational function if it is written as the sum of partial fractions

• Integrate each partial fraction separately

  •