Integration of Rational Functions

With rational integrals, it is assumed that the degree of the numerator is less than the denominator.

Integration of Rational Functions

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 Rational Function can be written as:

Rational Integrals with Simple Real Roots

The coefficients A, B and C are numbers that are obtained by performing the sum, identifying the coefficients and giving values to x.

Example

Rational Integrals with Simple Real Roots

Integration Operations

Integration Operations

Integration Operations

The sum is performed:

Integration Operations

Since the two fractions have the same denominator, the numerators must be equal:

Integration Operations

Calculate the coefficients of A, B and C by replacing x with the values that annul the denominator.

Rational Integrals with Simple Real Roots

Rational Integrals with Simple Real Roots

Rational Integrals with Simple Real Roots


2. Rational Integrals with Multiple Real Roots

The fraction Polynomial Fraction can be written as:

Rational Integrals with Multiple Real Roots

Example

Integral

Rational Integrals with Multiple Real Roots

Rational Integrals with Multiple Real Roots

Rational Integrals with Multiple Real Roots

To calculate the values of A, B and C, replace x with the values that annul the denominator.

Replace X

Integration Operations

Integration Operations

Integration Operations

Integration Operations


3. Rational Integrals with Simple Complex Roots

The fraction Polynomial Fraction can be written as:

Rational Integrals with Simple Complex Roots

This integral decomposes into a logarithmic and arctangent type function.

Example

Integral

Fractions

Integral Operations

Find the coefficients, realize the operations and equal the coefficients:

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Solution