# Integration of Rational Functions by Partial Fractions

1. The method of partial fractions is used to replace a
complex rational

function (in reduced form ) with a sum of simpler rational functions.

These simpler rational functions will be easier to integrate.

2. Let be a rational
function. Thus, P(x) and Q(x) are

polynomials.

3. The **method of partial fractions** lets us write

were p(x) is a polynomial and each F _{i}(x) is a
rational function of the

form:

or

The two rational expressions above are called **partial
fractions** and

is called the** partial fraction decomposition** of R(x).

4. **Method:** Find the partial fraction decomposition
of

a.** If the degree of P(x) is greater than or equal to
the degree
of Q(x),** use long division to divide P(x) by Q(x). (If the

degree of Q(x) is larger, then go to the next step.)

1. Be sure to express each polynomial P and Q in

descending powers of x .

2. The quotient ( result of the division ) is the polynomial p(x)

in the partial fraction decomposition.

3. The remainder has a degree less than the degree of the

devisor Q(t).

4. This will reduce R(x) to the form:

5. Proceed with the next steps on the rational expression

, which will have the numerator with a
smaller

degree than the denominator. (In the remainder of the

method, we will now refer to this rational expression as

P (x)/Q(x).)

b. **Factor the denominator **Q(x) into linear and quadratic

factors, possibly repeated.

1. Linear factors will have the form ax + b; or, if repeated

n times, the form .

2. Quadratic factors will have the form ax^{2} + bx + c, where

this quadratic polynomial cannot be factored further over

the real numbers . (b^{2} − 4ac < 0 ) If this factor is

repeated m times the form will be
.

3. This can always be done theoretically because any

polynomial of degree N has N real or complex roots,

possibly repeated. If the polynomial has real

coefficients, the real roots give the linear factors and the

complex roots give the quadratic factors (with real

coefficients).

c. The **linear factors** **of the
denominator **lead to

partial fractions of the form:

were are all constants
to be determined, some

of which could be zero. (There will be one set of n such

fractions for each linear factor of the form .)

d. The **quadratic factors**
**of the denominator**

lead to partial fractions of the form:

where and
are all constants to

be determined some of which could be zero. (Again one set of

m for each factor of the form
.)

e. The partial fraction decomposition of P(x)/Q(x) (assuming the

degree of Q(x) is larger than the degree of P(x)) is then the

**sum of all of the these partial fractions** related to all of the

linear and quadratic factors of the denominator Q (x).

f. There are two **method for determining the constants.**

Both methods start by **multiplying** both sides of the partial

fraction decomposition by **the denominator** of the original

rational function. This will clear all denominators. The

resulting equation (called the basic equation) will be **true for
all values of x .**

1. Since the basic equation is true for all values of x,

substitute a value for x that simplifies the equation. For

a distinct linear factor, pick x = the zero of this factor.

This will to eliminate most of the unknown constants in

the basic equation. Solve. Repeat for all distinct linear

factors.

2. Expand the basic equation and collect terms according to

their power of x. Equate coefficients of like powers of x

to obtain a system of linear equations involving the

unknown constants. Solve this system of equations.

3. A combination of these two methods is often the most

efficient method of finding these missing constants.

5. In Mathematica , a rational expression can be written in its partial

fraction decomposition using the command

**Apart.**

6. The rational function written in the form of its partial fraction

decomposition will be easier to integrate. For example:

For complete

the square in the denominator.

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