Partial Fractions

Definition

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.

5 Must Know Facts For Your Next Test

1. Partial fractions are 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 fractions involves factoring the denominator of the rational function and then expressing the function as a sum of simpler rational functions.
3. Partial fractions can be used to solve physical applications, such as finding the velocity or displacement of an object under the influence of a force that can be modeled as a rational function.
4. Trigonometric substitution is often used in conjunction with partial fractions to integrate rational functions that involve square roots in the denominator.
5. Separable differential equations, which can be solved by separating the variables, can sometimes be solved using partial fractions.

Review Questions

• Explain how partial fractions can be used to integrate rational functions that involve inverse trigonometric functions.
  • Partial fractions can be used to integrate rational functions that result in inverse trigonometric functions. By decomposing the rational function into simpler rational functions, the integration can be performed using the properties of inverse trigonometric functions. This allows for the evaluation of integrals that would otherwise be difficult or impossible to solve using other integration techniques.
• Describe the role of partial fractions in solving physical applications, such as finding the velocity or displacement of an object under the influence of a force that can be modeled as a rational function.
  • In physical applications, rational functions are often used to model forces or other quantities that depend on variables such as time or position. By using partial fractions to decompose these rational functions, it becomes possible to integrate them and find the velocity, displacement, or other physical quantities of interest. The decomposition into simpler rational functions allows for the application of integration techniques that may not be feasible for the original rational function.
• Analyze the relationship between partial fractions, trigonometric substitution, and the solution of separable differential equations.
  • Partial fractions, trigonometric substitution, and the solution of separable differential equations are closely related. Partial fractions can be used to integrate rational functions that involve square roots in the denominator, which can then be solved using trigonometric substitution. Additionally, separable differential equations, where the variables can be separated, can sometimes be solved using partial fractions. The decomposition of the rational function into simpler rational functions facilitates the integration and allows for the application of various techniques to find the solution to the differential equation.

Related terms

Rational Function: A rational function is a function that can be expressed as the ratio of two polynomials.

Decomposition: Decomposition refers to the process of breaking down a complex expression into simpler, more manageable components.

Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse of the standard trigonometric functions, such as sine, cosine, and tangent.