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.

- 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.

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