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

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

  3. Decompose the fraction into a sum of partial fractions;

  4. 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:
partial decomposition
The total number of undetermined coefficients Ai , Bi , Ki , Li , Mi , Ni , ... 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 Ai , Bi , Ki , Li , Mi , Ni , .... 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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