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: