## Rational functions: the development into partial fractions.

Recall that:

• A rational fraction is an expression of the form , where and are two polynomials.
• If , the rational fraction is proper; otherwise it is improper .
• An improper rational fraction is the sum of a polynomial and a proper rational fraction.
We decompose a rational fraction into a sum of partial fractions according to the following requirements:
• If the denominator of contains a factor of multiplicity 1, then there is a single fraction of the form in the desired developement ( is a constant).
• If the denominator of contains a factor of multiplicity , then the decomposition of into partial fractions contains a sum of fractions of the form , where the are constants.
• If the denominator of contains an irreducible factor of multiplicity 1, then there is a single fraction of the form in the desired developement ( and are constants).
• If the denominator of contains an irreducible factor of multiplicity , then the decomposition of into partial fractions contains a sum of fractions of the form , where the and are constants.

Example 8.2.10   Let , for . We have: . Therefore:

Example 8.2.11   Let , for . We have:

We compute separately the integral of each partial fraction; the first one and the third one use the techniques from  2.1, the second one requires the techniques of 2.2 (i.e. a substitution). Finally, we have:

Noah Dana-Picard 2007-12-28