10.7 The Method Of Partial Fractions
| 10.7 |
Return To Contents
Go To Problems & Solutions
Recall from algebra that a linear function is a polynomial of degree 1, ie a function of the form ax + b (its graph is a
line). A quadratic function is a polynomial of degree 2, ie a function of the form ax2 + bx + c. A real-valued polynomial is
said 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 to
a substitution method. If that’s the case, it’ll be expressed as a sum of simpler fractions, known as partial fractions, which
are easier to integrate.
Go To Problems & Solutions Return To Top Of Page
Consider, for example, the rational function:
Indeed it’s correct.
Another method of determining A and B is as follows. Multiplying both sides of Eq. [2.1] by the denominator x + 1 below
A we obtain:
Case 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 a
product of n distinct linear factors, say:
The constants Ai‘s, i = 1, 2, …, n, can be determined by the add-up-the-partial-fractions method or the limit-procedure
method as in the above example, where n = 2.
Go To Problems & Solutions Return To Top Of Page
| 3. Partial Fractions – Quadratic Factors |
Consider as another example the rational function:
Solving the system of equations:
Here, the limit-procedure method can be used to determine A, but there’s no simple way to use it to determine B or C.
Remark 3.1
You may ask why we don’t use a constant numerator for a partial fraction with a quadratic denominator to make things
simpler, 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 can be expanded to
A/(x + 2) + B/(x2 + x + 1). That’s the answer to the question.
Case 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 a
product of m distinct linear factors and n distinct irreducible quadratic factors, say:
Again corresponding to a linear denominator we use a constant numerator and corresponding to a quadratic denominator we use a linear numerator. That is, the degree of the numerator is less than that of the denominator by 1. The constants
Ai‘s, i = 1, 2, …, m, Bj‘s, and Cj‘s, j = 1, 2, …, n, can be determined by the add-up-the-partial-fractions method as in the
above example, where m = n = 1.
Go To Problems & Solutions Return To Top Of Page
| 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, with denominators having
exponents increasing from 1 to 4. There’s only 1 partial fraction corresponding to x – 3, and there are 3 corresponding to
x2 + 5, with denominators’ exponents increasing from 1 to 3.
The constants A1, A2, A3, A4, B, C1, C2, C3, D1, D2, and D3 can be determined by the add-up-the-partial-fractions
method.
Go To Problems & Solutions Return To Top Of Page
| 5. Partial Fractions – General Case |
The following theorem of polynomial algebra summarizes the general case of the partial-fraction expansion of a rational
function.
Theorem 5.1
| Let Q(x) be a polynomial. Then Q(x) can be factored into a product of a constant, linear factors, and irreducible quadratic factors, as follows: |
The proof of this theorem is omitted because it appropriately belongs to the domain of polynomial algebra. Here we simply
utilize the theorem.
Go To Problems & Solutions Return To Top Of Page
| 6. The Method Of Partial Fractions |
Example 6.1
Find:
Solution
EOS
Procedure
Suppose we are to find the integral:
If we don’t know how to do it, we decompose P(x)/Q(x) into a sum of partial fractions and integrate the resulting
expression. This technique is called the method of partial fractions. Its procedure is summarized as follows:
i. If the degree of P(x) is greater than or equal to that of Q(x), use polynomial long division to divide 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 the quotient, 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:
Return To Top Of Page
1. Calculate the following integrals.
Solution
Return To Top Of Page
2. Compute the following integrals.
Solution
Return To Top Of Page
3. Evaluate:
Solution
where C = (1/2)C1.
Return To Top Of Page
4. Find:
Solution
Return To Top Of Page
5. Calculate:
Solution
Let u = ex. Then du = ex dx = u dx, yielding dx = (du)/u. So:
Return To Top Of Page Return To Contents