Calculus/Integration techniques/Partial Fraction Decomposition
Suppose we want to find
. One way to do this is to simplify the integrand by finding constants A and B so that
This can be done by cross multiplying the fraction which gives
As both sides have the same denominator we must have
- 3x + 1 = A(x + 1) + Bx
This is an equation for x so it must hold whatever value x is. If we put in x = 0 we get 1 = A and putting x = − 1 gives − 2 = − B so B = 2. So we see that
Returning to the original integral
-

= 
= 
Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initial integral to a sum of simpler integrals. In fact this method works to integrate any rational function.
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[edit] Method of Partial Fractions
To decompose the rational function
:
1.1).
- Step 1 Use long division to ensure that the degree of P(x) is less than the degree of Q(x) (see Breaking up a rational function in section
- Step 2 Factor Q(x) as far as possible.
- Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.
To factor Q(x) we have to write it as a product of linear factors (of the form ax + b) and irreducible quadratic factors (of the form ax2 + bx + c with b2 − 4ac < 0).
Some of the factors could be repeated. For instance if Q(x) = x3 − 6x2 + 9x we factor Q(x) as
- Q(x) = x(x2 − 6x + 9) = x(x − 3)(x − 3) = x(x − 3)2.
It is important that in each quadratic factor we have b2 − 4ac < 0, otherwise it is possible to factor that quadratic piece further. For example if Q(x) = x3 − 3x2 − 2x then we can write
- Q(x) = x(x2 − 3x + 2) = x(x − 1)(x + 2)
We will now show how to write P(x) / Q(x) as a sum of terms of the form
and 
Exactly how to do this depends on the factorization of Q(x) and we now give four cases that can occur.
[edit] Q(x) is a product of linear factors with no repeats
This means that Q(x) = (a1x + b1)(a2x + b2)...(anx + bn) where no factor is repeated and no factor is a multiple of another.
For each linear term we write down something of the form
, so in total we write
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Example 1
Find Here we have P(x) = 1 + x2,Q(x) = (x + 3)(x + 5)(x + 7) and Q(x) is a product of linear factors. So we write
Multiply both sides by the denominator 1 + x2 = A(x + 5)(x + 7) + B(x + 3)(x + 7) + C(x + 3)(x + 5) Substitute in three values of x to get three equations for the unknown constants,
so A = 5 / 4,B = − 13 / 2,C = 25 / 4, and
We can now integrate the left hand side.
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[edit] Exercises
Evaluate the following by the method partial fraction decomposition.


[edit] Q(x) is a product of linear factors some of which are repeated
If (ax + b) appears in the factorisation of Q(x) k-times then instead of writing the piece
we use the more complicated expression

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Example 2
Find Here P(x) = 1 and Q(x) = (x + 1)(x + 2)2 We write
Multiply both sides by the denominator 1 = A(x + 2)2 + B(x + 1)(x + 2) + C(x + 1) Substitute in three values of x to get 3 equations for the unknown constants,
so A = 1, B = − 1, C = − 1, and
We can now integrate the left hand side. |
[edit] Exercise
using the method of partial fractions.
[edit] Q(x) contains some quadratic pieces which are not repeated
If (ax2 + bx + c) appears we use 
[edit] Exercises
Evaluate the following using the method of partial fractions.


[edit] Q(x) contains some repeated quadratic factors
If (ax2 + bx + c) appears k-times then use
[edit] Exercise
Evaluate the following using the method of partial fractions.







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