Integration of Rational Functions

A rational function notation of rational function

, where P(x) and Q(x) are both polynomials, can be integrated in four steps:

Reduce the fraction if it is improper (i.e. degree of P(x) is greater than degree of Q(x));

Factor Q(x) into linear and/or quadratic (irreducible) factors;

Decompose the fraction into a sum of partial fractions;

Calculate integrals of each partial fraction.

Consider the specified steps in more details.

Step 1. Reducing an improper fraction

If the fraction is improper (i.e. degree of P(x) is greater than degree of Q(x)), divide the numerator P(x) by the denominator Q(x) to obtain

$reducing an improper fraction$

where

is a proper fraction.

Step 2. Factoring Q(x) into linear and/or quadratic factors

Write the denominator Q(x) as

where quadratic functions are irreducible, i.e. do not have real roots.

Step 3. Decomposing the rational fraction into a sum of partial fractions.

Write the function as follows:

The total number of undetermined coefficients A_i , B_i , K_i , L_i , M_i , N_i , ... must be equal to the degree of the denominator Q(x).

Then equate the coefficients of equal powers of x by multiplying both sides of the latter expression by Q(x) and write the system of linear equations in A_i , B_i , K_i , L_i , M_i , N_i , .... The resulting system must always have a unique solution.

Step 4. Integrating partial fractions.

Use the following 6 formulas to evaluate integrals of partial fractions with linear and quadratic denominators:

For fractions with quadratic denominators, first complete the square:

where

Then use the formulas:

The integral

can be calculated in k steps using the reduction formula:

Example 1

Evaluate the integral

Solution.

Decompose the integrand into partial functions:

Equate coefficients:

Hence,

Then

The integral is equal to

Example 2

Evaluate

Solution.

First we divide the numerator by the denominator, obtaining

Then

Example 3

Evaluate the integral

Solution.

We can write:

Example 4

Evaluate the integral

Solution.

Decompose the integrand into partial functions:

Equate coefficients:

Hence,

Then

The integral is equal to

Example 5

Evaluate

Solution.

Decompose the integrand into the sum of two fractions:

Equate coefficients:

Hence

The integrand can be written as

The initial integral becomes

Example 6

Find the integral

Solution.

We can factor the denominator in the integrand:

Decompose the integrand into partial functions:

Equate coefficients:

Hence,

Then

Now we can calculate the initial integral:

Example 7

Calculate the integral

Solution.

Rewrite the denominator in the integrand as follows:

The factors in the denominator are irreducible quadratic factors since they have no real roots. Then

Equate coefficients:

This yields

Hence,

Integrating term by term, we obtain the answer:

We can simplify this answer. Let

Then

Hence,

. The complete answer is

Example 8

Evaluate the integral

Solution.

We can factor the denominator in the integrand:

Decompose the integrand into partial functions:

Equate coefficients:

Hence,

Thus, the integrand becomes

So, the complete answer is

Example 9

Calculate the integral

Solution.

Decompose the integrand into partial functions, taking into account that the denominator has a third degree root:

Equate coefficients:

We get the following system of equations:

Hence,

The initial integral is equal to

Example 10

Find the integral

Solution.

Since

is reducible, we complete the square in the denominator:

Now, we can compute the integral using the reduction formula

Then


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