Lecture 29
18.01 Fall 2006
Lecture
29:
Partial
Fractions
We continue the discussion we started last lecture about integrating rational functions.
We
defined a rational function as the ratio of two polynomials:
P
(
x
)
Q
(
x
)
We looked at the example
1
3
x

1
+
x
+ 2
dx
=
ln

x

1

+
3 ln

x
+ 2

+
c
That same problem can be disguised:
1
+
3
=
(
x
+
2)
+
3(
x

1)
=
4
x

1
x

1
x
+ 2
(
x

1)(
x
+
2)
x
2
+
x

2
which leaves us to integrate this:
4
x

1
dx
=
???
x
2
+
x

2
P
(
x
)
Goal
: we want to figure out a systematic way to split
into simpler pieces.
Q
(
x
)
First, we factor the denominator
Q
(
x
)
.
4
x

1
=
4
x

1
=
A
+
B
x
2
+
x

2
(
x

1)(
x
+
2)
x

1
x
+ 2
There’s a slow way to find
A
and
B
. You can clear the denominator by multiplying through by
(
x

1)(
x
+
2)
:
(4
x

1)
=
A
(
x
+
2)
+
B
(
x

1)
From this, you find
4 =
A
+
B
and

1 = 2
A

B
You can then solve these simultaneous linear equations for
A
and
B
. This approach can take a very
long time if you’re working with 3, 4, or more variables.
There’s a faster way, which we call the “coverup method”. Multiply both sides by
(
x

1)
:
4
x

1
B
x
+ 2
=
A
+
x
+
2
(
x

1)
Set
x
= 1
to make the
B
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 Fall '08
 Staff
 Polynomials, Fractions, Rational Functions, Fraction, Rational function, dx, x2 dx, coverup method

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