Download Lecture Notes for Section 6.4

Calc 2 Lecture Notes
Section 6.4
Page 1 of 4
Section 6.4: Integration of Rational Functions Using Partial Fractions
Big idea: In this section, we’ll examine a technique called partial fraction decomposition that is
used for evaluating the integral of a rational function (i.e., the ratio of two polynomials).
Big skill:. You should be able to decompose various types of rational functions into simpler
fractions that can be easily integrated using known formulas.
I. Warm-up: Adding rational functions.
2
3

1. Combine
2x  5 x  9
2  x  9   3  2 x  5
2
3


2x  5 x  9
 2 x  5 x  9 

2 x  18  6 x  15
 2 x  5 x  9 
8x  3
2 x  13x  14
2. Our goal in this section is to figure out how to start with an integrand looking
8x  3
2
3

like 2
and convert it to
:
2 x  13 x  14
2x  5 x  9
8x  3
2
3
 2 x 2  13x  14 dx   2 x  5 dx   x  9 dx  ln 2 x  5  3ln x  9  c

2
II. Basic example: Linear polynomial in the numerator and a factorizable quadratic
a x  b1
a1 x  b1
polynomial in the denominator: 21
.

ax  bx  c  a2 x  b2  a3 x  b3 
1. A rational function of this form can always be written as:
a1 x  b1
A
B
.


 a2 x  b2  a3 x  b3  a2 x  b2 a3 x  b3
i.
ii.
iii.
iv.
v.
Multiply both sides by  a2 x  b2  a3 x  b3  .
Expand the right-hand side.
Equate like terms to get a system of linear equations in A and B.
Solve the system for A and B.
Shortcut: a1 x  b1  A  a3 x  b3   B  a2 x  b2  ; plug in the zeros of each
factor to get A and B.
Calc 2 Lecture Notes
2. Practice:
x
Section 6.4
Page 2 of 4
3x  2
dx 
 x6
2
III. Partial Fraction Decomposition for rational functions with distinct linear factors: If
a rational function has n distinct linear factors (i.e., each factor is different) in the
denominator and a numerator polynomial P(x) that is of degree less than n, then that
function can be decomposed as:
P  x
cn
c1
c2


 
an x  bn
 a1 x  b1  a2 x  b2   an x  bn  a1 x  b1 a2 x  b2
for constants c1, c2, …, cn.
2 x2  1
1. Practice:  3
dx 
3x  x 2  2 x
2. To check your work using a graphing calculator (with no CAS), or Winplot:
Graph the integrand and the derivative of the antiderivative on the same screen…
Y1 = (2X2+1)/(3X3+X2-2X)
Y2=nDeriv(-0.5ln(X)+0.6ln(X+1)+1.7/3*ln(3X-2),X,X)
… or use the derive button on the inventory box in winplot…
3. Caveat: The degree of the numerator must be less than the degree of the
denominator for this technique to work  Do long division of the polynomials
before integrating if this is not the case.
Calc 2 Lecture Notes
4. Practice:
Section 6.4
Page 3 of 4
x3  10 x  1
 x2  5x  6 dx 
IV. Partial Fraction Decomposition for rational functions with repeated linear factors:
If a rational function has a denominator with a single linear factor of multiplicity n > 1
and a numerator polynomial P(x) that is of degree less than n, then that function can be
decomposed as:
P  x
cn
c
c2
 1 
 
n
2
n
 ax  b  ax  b  ax  b 
 ax  b 
for constants c1, c2, …, cn.
x 3
dx 
1. Practice:  2
x  2x 1
x2  1
dx 
2. Practice:  3
x  2 x2  x
Calc 2 Lecture Notes
Section 6.4
Page 4 of 4
V. Partial Fraction Decomposition for rational functions with distinct irreducible
quadratic factors: If a rational function has n distinct irreducible quadratic factors in the
denominator and a numerator polynomial P(x) that is of degree less than 2n, then that
function can be decomposed as:
P  x
a x
1
2
 b1 x  c1  a2 x 2  b2 x  c2 
a x
n
2
 bn x  cn 
An x  Bn
A1 x  B1
A2 x  B2

 
2
2
a1 x  b1 x  c1 a2 x  b2 x  c2
an x 2  bn x  cn
for constants A1, A2, …, An, B1, B2, …, Bn.,
2 x2  5x  2
1. Practice: 
dx 
x3  x

VI. What do you think happens for a rational function with repeated irreducible
quadratic factors?
x5  x 4  x3  x 2  x  1
1. Practice: 
dx 
2
2
 x  1  x 2  x  1