# Integration of Rational Functions

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: