Download Techniques of Integration: Partial Fraction Decomposition (sec 7.5)

Warm-Up: Common Denominators
Notes
Write the sum of rational functions as a single rational
function, by finding a common denominator:
x +1
2
+
=?
x2 + 4 x + 3
Notes
Techniques of Integration:
Partial Fraction Decomposition
(sec 7.5)
22 September 2014
Today, We Will:
I
Learn to apply the method of partial fraction
decomposition to re-write a rational function as a sum of
rational functions whose denominators are polynomials of
degree less than or equal to two.
Notes
Terminology
Notes
Recall:
I
A rational function is a function that is the quotient of
two polynomials.
I
The degree of a polynomial is the highest power of a
variable occurring in the polynomial.
I
A polynomial of degree one is called a linear polynomial.
I
A polynomial of degree two is called a quadratic
polynomial.
I
A quadratic polynomial that cannot be factored into
linear polynomials is called irreducible .
Motivation
Notes
Consider the integral:
Z
3x 2 + 4x + 11
dx.
x 3 + 3x 2 + 4x + 12
Problem: Except in rare situations where a u-substitution is
possible, we don’t have any integration techniques that let us
deal with rational functions if the degree of the polynomial in
the denominator is greater than two.
continued . . .
Motivation, Continued
Notes
From the warm-up, however, we know that:
Z
3x 2 + 4x + 11
dx =
x 3 + 3x 2 + 4x + 12
=
These we can integrate!
Z
x +1
2
+
dx
x2 + 4 x + 3
Z
x
dx +
x2 + 4
Z
1
dx +
x2 + 4
Z
2
dx
x +3
Partial Fraction Decomposition
Notes
Partial fraction decomposition is an algebraic technique that
“undoes” the process of combining rational functions by
finding a common denominator.
It allows us to take a rational function whose denominator is a
polynomial of degree greater than two, and write it as a sum
of rational functions whose denominators are linear and
quadratic polynomials.
Once we have found the decomposition of the original rational
function, we can apply a combination of techniques — usually
u-substitution and trigonometric substitution — to evaluate
the integral.
Examples: Partial Fraction Decomposition
I
R(x) =
x +2
x2 + x − 6
I
R(x) =
x +2
(x + 3)(x − 2)2
The Method
Notes
Notes
I
Make sure the degree of the denominator is greater than
the degree of the numerator. If it is not, divide the
numerator by the denominator to re-write the original
function as the sum of a polynomial and a rational
function.
I
Factor the denominator into powers of linear and
irreducible quadratic polynomials.
continued . . .
The Method, Continued
I
Notes
For every power of a linear factor (ax + b)n , write the
sum:
A1
A2
An
+
+ ··· +
ax + b (ax + b)2
(ax + b)n
where A1 , A2 , . . . , A3 are unknown constants.
continued . . .
The Method, Continued
I
Notes
For every power of an irreducible quadratic factor
(ax 2 + bx + c)n , write the sum:
A1 x + B1
A2 x + B2
An x + Bn
+···+
+
ax 2 + bx + c (ax 2 + bx + c)2
(ax 2 + bx + c)n
where A1 , A2 , . . . , An and B1 , B2 , . . . , Bn are unknown
constants.
continued . . .
The Method, Continued
I
Combine all resulting sums into a single rational function.
I
Clear the denominators and equate the numerator of the
resulting rational function and the numerator of the
original rational function.
I
Solve for the unknown constants.
Notes