Integration With Partial Fractions
MAT1242: Calculus II
University of Eswatini.
Integration with partial fractions is a powerful technique used to simplify and evaluate integrals of
rational functions. It allows us to break down complex rational expressions into simpler fractions, making
integration more manageable. In this comprehensive guide, we’ll cover the following topics:
1. Introduction to Partial Fractions
2. Decomposition of Rational Functions
3. Types of Partial Fractions
4. Step-by-Step Examples
1. Introduction to Partial Fractions
Partial fractions come into play when we encounter rational functions (quotients of polynomials). The
idea is to express a complicated rational function as a sum of simpler fractions. This decomposition
allows us to integrate each term separately.
2. Decomposition of Rational Functions
Given a rational function:
where (P(x)) and (Q(x)) are polynomials, we follow these steps to decompose it into partial fractions:
1. Factor the denominator (Q(x)): Express (Q(x)) as a product of irreducible (non-factorable)
polynomials.
2. Write the partial fraction form:
where are the irreducible factors of (Q(x)), and are constants.
3. Find the constants : Multiply both sides by (Q(x)) and equate coefficients of corresponding
powers of (x).
MAT1242: Calculus II
University of Eswatini.
Integration with partial fractions is a powerful technique used to simplify and evaluate integrals of
rational functions. It allows us to break down complex rational expressions into simpler fractions, making
integration more manageable. In this comprehensive guide, we’ll cover the following topics:
1. Introduction to Partial Fractions
2. Decomposition of Rational Functions
3. Types of Partial Fractions
4. Step-by-Step Examples
1. Introduction to Partial Fractions
Partial fractions come into play when we encounter rational functions (quotients of polynomials). The
idea is to express a complicated rational function as a sum of simpler fractions. This decomposition
allows us to integrate each term separately.
2. Decomposition of Rational Functions
Given a rational function:
where (P(x)) and (Q(x)) are polynomials, we follow these steps to decompose it into partial fractions:
1. Factor the denominator (Q(x)): Express (Q(x)) as a product of irreducible (non-factorable)
polynomials.
2. Write the partial fraction form:
where are the irreducible factors of (Q(x)), and are constants.
3. Find the constants : Multiply both sides by (Q(x)) and equate coefficients of corresponding
powers of (x).
