Partial Fraction Expansion

Definition

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.

5 Must Know Facts For Your Next Test

1. Partial fraction expansion is used to integrate rational functions that cannot be easily integrated using other methods, such as substitution or integration by parts.
2. The process of partial fraction expansion involves identifying the factors of the denominator and then expressing the original function as a sum of simpler fractions.
3. Partial fraction expansion is particularly useful when dealing with repeated linear factors or quadratic factors in the denominator of a rational function.
4. The specific steps involved in partial fraction expansion depend on the nature of the denominator, such as the presence of distinct linear factors, repeated linear factors, or irreducible quadratic factors.
5. Partial fraction expansion can also be used to evaluate definite integrals involving rational functions, as the simpler fractions obtained can be integrated more easily.

Review Questions

• Explain the purpose and importance of partial fraction expansion in calculus.
  • The primary purpose of partial fraction expansion is to decompose a complex rational function into a sum of simpler rational functions, which can then be more easily integrated or evaluated. This technique is particularly useful when dealing with rational functions that cannot be integrated using other methods, such as substitution or integration by parts. Partial fraction expansion is an essential tool in calculus, as it allows for the evaluation of integrals involving rational functions and the solution of differential equations that involve rational functions.
• Describe the general steps involved in the process of partial fraction expansion.
  • The process of partial fraction expansion typically involves the following steps: 1) Identify the factors of the denominator of the rational function, which may include distinct linear factors, repeated linear factors, or irreducible quadratic factors. 2) Determine the appropriate form of the partial fraction decomposition based on the nature of the denominator. 3) Set up a system of equations using the coefficients of the partial fraction decomposition and solve for the unknown coefficients. 4) Express the original rational function as the sum of the simpler rational functions obtained through the partial fraction expansion.
• Explain how partial fraction expansion can be used to evaluate definite integrals involving rational functions.
  • Partial fraction expansion is a valuable tool for evaluating definite integrals involving rational functions. By decomposing the original rational function into a sum of simpler rational functions, the integration process becomes much more manageable. The simpler fractions obtained through partial fraction expansion can then be integrated individually using standard integration techniques, such as the power rule, the logarithm rule, or the inverse tangent rule. The final result is the sum of the integrals of the individual partial fractions, which represents the definite integral of the original rational function.

Related terms

Rational Function: A rational function is a function that can be expressed as the quotient of two polynomial functions.

Proper Fraction: A proper fraction is a rational function where the degree of the numerator is less than the degree of the denominator.

Improper Fraction: An improper fraction is a rational function where the degree of the numerator is greater than or equal to the degree of the denominator.