Phong Do
Integration using Partial Fraction
Partial fraction is a method allowing you to take a large fraction and “decompose” it into several single fractions.
Working with these single fractions is often easier as the below example will show.
Let’s say you have this
rational expression
:
1
(
2)(
2)
x
x
+
−
From Pre-Cal (and if you forgot, we will do a quick review on Partial Fraction Decomposition) you can decompose
this into 2 single fractions as:
1/ 4
1/ 4
1
(
2)(
2)
2
2
x
x
x
x
−
=
+
+
−
+
−
If we take the above example and turn in into an integral, you will see how partial fraction will help you:
1/ 4
1/ 4
(
2)(
2)
2
2
dx
dx
dx
x
x
x
x
−
=
+
+
−
+
−
∫
∫
∫
The 2 integrals containing the single fractions can be integrated as:
1
1
ln
2
ln
2
4
4
x
x
C
−
+
+
−
+
So as you can see, by splitting a single fraction into 2 smaller fractions each with single denominator, integration
can be conveniently carried out.
Proper
and
Improper
Rational Functions: A Quick Pre-Cal Review
Rational Functions has the form:
( )
( )
P x
Q x
And in Algebra, you might have recalled that:
•
A
proper
rational function is one where the
power of the numerator is less than the power of the
denominator
•
An
improper
rational function is one where the
power of the numerator is greater than or equal to the
power of the denominator
.
•
Partial Fraction technique requres a rational function to be
PROPER
.
3
Proper
(
2)(
4)
x
x
x
⇒
−
+
3
4
2
2
3
Proper
3
5
x
x
x
x
−
⇒
+
−
3
2
2
3
Improper
4
7
1
x
x
x
x
⇒
+
+
+
+
•
If a rational expression is improper, you use polynomial long division (or synthetic division if appropriate)
to reduce it to a
polynomial + a proper rational function
(
)
3
2
2
2
3
4
7
3
2
3
4
1
1
x
x
x
x
x
x
x
⇒
+
+
+
−
+
+
+
+
The
2
3
2
1
x
x
−
+
is proper which you can use Partial Fraction on.

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