L ectur e1 7| 1 Integration by partial fractions
To integrate the fraction
one cannot use all the previous methods (substitution, by parts).
However, one observe that the fraction can be reduced into
i.e. it is the sum of other fractions. Then
L ectur e1 7| 2 In this section we study the technique to integrate fractions
We will see that the fraction
can be reduced into sum (or difference) of other fractions called partial fractions.
This is called the partial fractions expansion of .
Then the integral
is the sum of the integrals of partial fractions.
L ectur e1 7| 3 EX
L ectur e1 7| 4 If we factorize
we call
In the case (or ), the term is called simple.
If (or ), the term is said to be repeated with order (resp., ).
L ectur e1 7| 5 EX The polynomial
has
a simple linear term ,
a repeated linear term , order a repeated quad term ,
order 2
L ectur e1 7| 6 Partial Fraction (with Long division)
If , using long division we can express
where . So
can be easily integrated.
If then . So long division is not required.
L ectur e1 7| 7 EX Use long division to reduce
Then integrate
L ectur e1 7| 8 Now we have to study
L ectur e1 7| 9 Partial Fraction: (simple linear terms) If
where are distinct real numbers, then there are such that
can be computed from
Formula
L ectur e1 7| 10 EX Find the partial fraction expansion of
Then integrate
L ectur e1 7| 11 EX Find the partial fraction expansion of
Then evaluate
L ectur e1 7| 12 Generally, if the factorization of has a simple linear term (together with others, which could be repeated, or quadratics), this term gives rise to
in the partial fraction expansion of
is computed by the same formula:
L ectur e1 7| 13 Partial Fraction: (repeated linear term)
If the factorization of has
then this term gives rise to
in the partial fraction expansion of
. can be computed from
Formula
L ectur e1 7| 14 EX Find the partial fraction expansion of
Then evaluate the integral
L ectur e1 7| 15 EX Evaluate
L ectur e1 7| 16 Partial Fraction: (simple quad terms)
If the factorization of has a simple quad term
then this term gives rise to
in the partial fraction expansion of
. Formula
Repeated quad terms need the trig.
subst., will be discussed later.
L ectur e1 7| 17 EX Find the partial fraction expansion of
Then evaluate the integral
L ectur e1 7| 18 EX (Change of variable first) Evaluate
L ectur e1 7| 19 EX (Change of variable first) Evaluate
