With rational integrals, it is assumed that the degree of the numerator is less than the denominator.
Once it is known that the denominator has a higher degree than the numerator, decompose the denominator into factors.
Depending on the roots of the denominator, find the following types of rational integrals:
1. Rational Integrals with Simple Real Roots
The fraction 
can be written as:
The coefficients A, B and C are numbers that are obtained by performing the sum, identifying the coefficients and giving values to x.
Example
The sum is performed:
Since the two fractions have the same denominator, the numerators must be equal:
Calculate the coefficients of A, B and C by replacing x with the values that annul the denominator.
2. Rational Integrals with Multiple Real Roots
The fraction can be written as:
Example
To calculate the values of A, B and C, replace x with the values that annul the denominator.
3. Rational Integrals with Simple Complex Roots
The fraction can be written as:
This integral decomposes into a logarithmic and arctangent type function.
Example
Find the coefficients, realize the operations and equal the coefficients:
