Calculus/Integration techniques/Partial Fraction Decomposition
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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 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.
Method of Partial Fractions:
- Step 1 Use long division to ensure that the degree of P(x) less than the degree of Q(x).
- 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.
Case (a) 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
[edit] 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.

Case (b) 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

[edit] 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. 
Case (c) Q(x) contains some quadratic pieces which are not repeated.
If (ax2 + bx + c) appears we use 
Case (d) Q(x) contains some repeated quadratic factors.
If (ax2 + bx + c) appears k-times then use



