the method of partial fractions

8/3/2019 The Method of Partial Fractions

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The Method Of Partial Fractions

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1. Notes

Recall from algebra that a linear function is a polynomial of degree 1, ie a function of the form ax + b (itsgraph is aline ). A quadratic function is a polynomial of degree 2, ie a function of the form ax 2 + bx + c. A real-valued polynomial issaid to be irreducible if it can't be factored. Note that all real-valued linear functions are irreducible.

A rational function is a ratio or fraction P ( x) / Q( x) where P ( x) and Q( x) are polynomials.

In this section we're concerned with the integration of rational functions. A rational function may not readily lend itself toa substitution method. If that's the case, it'll be expressed as a sum of simpler fractions, known as partialfractions, whichare easier to integrate.

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2 . Partial Fractions Linear Factors

Consider, for example, the rational function:


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Indeed it's correct.

Another method of determining A and B is as follows. Multiplying both sides of Eq. [2.1] by thedenominator x + 1 below A we obtain:

C ase Of n Distinct Linear Factors

In general, if the degree of the numerator P ( x) is less than that of the denominator Q( x) and if Q( x)factors into aproduct of n distinct linear factors, say:


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Solving the system of equations:


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Here, the limit-procedure method can be used to determine A, but there's no simple way to use it todetermine B or C .

R emark 3 .1

You may ask why we don't use a constant numerator for a partial fraction with a quadratic denominatorto make thingssimpler, like this: A / ( x + 2) + B / ( x2 + x + 1). Well, let's see:

which has no solutions. That is, There are no constants A and B such that the given rational function canbe expanded to A / ( x + 2) + B / ( x2 + x + 1). That's the answer to the question.

C ase Of m Distinct Linear Factors And n Distinct Quadratic Factors

In general, if the degree of the numerator P ( x) is less than that of the denominator Q( x) and if Q( x)factors into aproduct of

mdistinct linear factors and

ndistinct irreducible quadratic factors, say:

Again corresponding to a linear denominator we use a constant numerator and corresponding to aquadratic denominatorwe use a linear numerator. That is, the degree of the numerator is less than that of the denominator by1. The constants Ai's, i = 1, 2, ..., m, B j's, and C j's, j = 1, 2, ..., n, can be determined by the add-up-the-partial-fractionsmethod as in theabove example, where m = n = 1.



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4 . Partial Fractions Multiplicity

Consider, for example, the rational function:

Since the multiplicity of the factor x is 4, there are 4 partial fractions corresponding to x, withdenominators havingexponents increasing from 1 to 4. There's only 1 partial fraction corresponding to x 3, and there are 3corresponding to x2 + 5, with denominators' exponents increasing from 1 to 3.

The constants A1, A2, A3, A4, B, C 1, C 2, C 3, D1, D2, and D3 can be determined by the add-up-the-partial-fractionsmethod.


5 . Partial Fractions General C ase

The following theorem of polynomial algebra summarizes the general case of the partial-fractionexpansion of a rationalfunction.

Theorem 5 .1

Let Q( x) be a polynomial. Then Q( x) can be factored into a product of a constant, linear factors, and irreduciblequadratic factors, as follows:


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The proof of this theorem is omitted because it appropriately belongs to the domain of polynomialalgebra. Here we simplyutilize the theorem.


6 . The Method Of Partial Fractions

Example 6 .1

Find:


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i. If the degree of P ( x) is greater than or equal to that of Q( x), use polynomial long division todivide P ( x) by Q( x) to

obtain P ( x) / Q( x) = q( x) + R( x) / Q( x) (from P ( x) = q( x)Q( x) + R( x)), where q( x) is thequotient, R( x) is the

remainder, and the degree of R( x) is less than that of Q( x).

ii. Factor the denominator Q( x) into linear and/or irreducible quadratic factors.

iii. Perform the partial-fraction expansion on P ( x) / Q( x), or on R( x) / Q( x) if part i is carried out.

iv. Integrate the resulting expression of P ( x) / Q( x).

Note On Long Division

For example, given:

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Problems & S olutions


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1. Calculate the following integrals.

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2 . Compute the following integrals.

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3 . Evaluate:

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where C = (1/2) C 1.


4 . Find:

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Let u = e x. Then d u = e x dx = u dx, yielding dx = ( d u )/ u . So:

the method of partial fractions

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