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3.4 Partial Fractions

3 min read•june 24, 2024

is a powerful technique for simplifying complex . By breaking down these functions into simpler parts, it makes integration much easier and more manageable.

This method involves factoring denominators and expressing functions as sums of . It's especially useful for integrating rational functions with linear, repeated, or in the denominator.

Partial Fraction Decomposition

Partial fractions for integration

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  • Simplify complex rational functions into a sum of simpler rational functions
  • Makes integration easier by breaking down the original function
  • Factor the denominator and express the function as a sum of partial fractions
    • Each partial fraction corresponds to a factor in the denominator
  • General form: P(x)Q(x)=A1(x−a1)+A2(x−a2)+...+An(x−an)\frac{P(x)}{Q(x)} = \frac{A_1}{(x-a_1)} + \frac{A_2}{(x-a_2)} + ... + \frac{A_n}{(x-a_n)}
    • P(x)P(x) and Q(x)Q(x) are polynomials
    • a1,a2,...,ana_1, a_2, ..., a_n are the roots of Q(x)Q(x) (zeros of the denominator)
    • A1,A2,...,AnA_1, A2, ..., A_n are constants to be determined
  • Find the constants by multiplying both sides by Q(x)Q(x) and solving the resulting system of linear equations
  • Examples: 2x+1x2−x−2\frac{2x+1}{x^2-x-2}, 3x−2(x−1)(x+2)\frac{3x-2}{(x-1)(x+2)}

Simple linear factors in functions

  • Denominator contains factors of the form (x−a)(x-a), where aa is a constant
  • Each simple linear factor (x−a)(x-a) corresponds to a partial fraction of the form Ax−a\frac{A}{x-a}
    • AA is a constant to be determined
  • Find the constant AA by multiplying both sides by (x−a)(x-a) and evaluating at x=ax=a
    • Eliminates all other terms, leaving only the constant AA
  • Examples: 2x+1(x−1)(x+3)\frac{2x+1}{(x-1)(x+3)}, 3(x+2)(x−4)\frac{3}{(x+2)(x-4)}

Integration with repeated linear factors

  • Denominator contains factors of the form (x−a)n(x-a)^n, where aa is a constant and nn is a positive integer
  • Each repeated linear factor (x−a)n(x-a)^n corresponds to partial fractions of the form:
    • A1x−a+A2(x−a)2+...+An(x−a)n\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}
    • A1,A2,...,AnA_1, A_2, ..., A_n are constants to be determined
  • Find the constants by multiplying both sides by (x−a)n(x-a)^n and evaluating the derivatives at x=ax=a
    • Creates a system of linear equations to solve for the constants
  • Examples: x+2(x−1)3\frac{x+2}{(x-1)^3}, 3x−1(x+2)2(x−3)\frac{3x-1}{(x+2)^2(x-3)}

Quadratic factors in partial fractions

  • Denominator contains factors of the form (ax2+bx+c)(ax^2+bx+c), where aa, bb, and cc are constants
    • Quadratic expression is irreducible (cannot be factored further)
  • Each quadratic factor (ax2+bx+c)(ax^2+bx+c) corresponds to a partial fraction of the form Ax+Bax2+bx+c\frac{Ax+B}{ax^2+bx+c}
    • AA and BB are constants to be determined
  • Find the constants AA and BB by multiplying both sides by (ax2+bx+c)(ax^2+bx+c) and equating coefficients of like terms
    • Creates a system of linear equations to solve for the constants
  • After decomposition, integrate each partial fraction separately using appropriate techniques
  • Examples: 2x+1x2+4x+3\frac{2x+1}{x^2+4x+3}, 3x−2(x2−1)(x+2)\frac{3x-2}{(x^2-1)(x+2)}

Advanced Techniques for Partial Fraction Decomposition

  • : Rational expressions that can be decomposed into simpler fractions
  • : A method used to equate the numerator of the original fraction with the sum of partial fractions
  • : A shortcut method for dividing polynomials, useful in preparing fractions for decomposition
  • : When the denominator contains factors with complex roots, leading to partial fractions with complex coefficients
  • : The process of breaking down a complex fraction into simpler terms for easier integration

Key Terms to Review (22)

Algebraic Fractions: Algebraic fractions are mathematical expressions that represent a ratio of two algebraic expressions, where the numerator and denominator can contain variables, coefficients, and operations. They are a fundamental concept in algebra and calculus, often encountered when working with rational functions and partial fractions.
Complex Roots: Complex roots refer to the solutions to polynomial equations that involve imaginary numbers. These roots occur in complex conjugate pairs and are an essential concept in the study of partial fractions.
Cover-up Method: The cover-up method is a technique used in the context of partial fractions to simplify the integration of rational functions. It involves manipulating the denominator of the original expression to create a new expression that can be more easily integrated.
Cross Multiplication: Cross multiplication is a technique used to solve proportions or ratios by multiplying the numerators and denominators of the two fractions in a specific way. It allows for the comparison and equating of two ratios to find an unknown value.
Distinct Linear Factors: Distinct linear factors refer to the unique linear expressions that arise when decomposing a rational function into partial fractions. These linear factors represent the denominators of the partial fraction components and are essential in the process of partial fraction expansion.
Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex number coefficients has at least one complex number solution. This theorem is a fundamental result in abstract algebra and complex analysis, connecting the properties of polynomials to the structure of the complex number system.
Heaviside Method: The Heaviside method, also known as the method of partial fractions, is a technique used to express a rational function as a sum of simpler rational functions. This method is particularly useful in the context of solving differential equations and Laplace transforms.
Improper Rational Function: An improper rational function is a rational function where the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial. This type of function can be further decomposed using the technique of partial fractions to simplify its expression and make it easier to evaluate.
Irreducible Quadratic Factors: Irreducible quadratic factors are polynomial factors of a quadratic expression that cannot be further factored into smaller polynomial factors. They represent the simplest form of a quadratic expression that cannot be broken down any further.
Long Division: Long division is a method for dividing a large number by another number, typically by breaking down the division process into a series of smaller steps. It is a systematic approach to division that allows for the calculation of quotients and remainders, even for complex numerical problems.
Method of equating coefficients: The method of equating coefficients is a technique used to determine the values of unknown constants in an algebraic expression by matching corresponding coefficients on both sides of an equation. This method is often employed in the context of integrating rational functions using partial fractions.
Method of strategic substitution: Method of strategic substitution involves replacing a complex expression with a simpler one to facilitate integration. This technique often simplifies the problem, making it easier to solve.
Partial fraction decomposition: Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This is particularly useful for integrating rational functions.
Partial Fraction Decomposition: Partial fraction decomposition is a technique used in calculus to express a rational function as a sum of simpler rational functions. This method is particularly useful in the context of evaluating integrals involving rational functions and in the study of infinite series.
Partial Fraction Expansion: Partial fraction expansion is a technique used in calculus to decompose a rational function into a sum of simpler rational functions. This process involves breaking down a complex fraction into a combination of more manageable fractions, which can then be integrated or evaluated more easily.
Partial Fractions: Partial fractions is a technique used to decompose a rational function into a sum of simpler rational functions. This method is often employed when integrating rational functions, as it allows for the use of inverse trigonometric functions, integration by parts, and other integration techniques.
Polynomial Division: Polynomial division is a mathematical operation that involves dividing a polynomial by another polynomial, similar to how integers are divided. The goal of polynomial division is to find the quotient and the remainder when one polynomial is divided by another.
Proper Rational Function: A proper rational function is a rational function in which the degree of the numerator is less than the degree of the denominator. This type of function is commonly encountered in the context of partial fractions, where it is used to represent and analyze complex rational expressions.
Quadratic Factors: Quadratic factors refer to the factors of a quadratic expression, which is a polynomial equation of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. These factors are the values of $x$ that make the quadratic expression equal to zero, and they are essential in understanding the behavior and properties of quadratic functions.
Rational functions: Rational functions are ratios of two polynomials. They are often used in calculus to perform integration using partial fractions.
Repeated Linear Factors: Repeated linear factors refer to the occurrence of linear factors that appear more than once in the denominator of a rational function. These repeated factors are essential in the process of decomposing the function into a sum of partial fractions, which is a fundamental technique in the study of calculus.
Synthetic Division: Synthetic division is a shortcut method for dividing a polynomial by a linear expression of the form $(x - a)$. It allows for the efficient computation of the quotient and remainder without having to perform long division.
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3.5 Other Strategies for Integration