Problem 1 - Calculus Early Transcendentals

Question

What kinds of functions can be integrated using partial fraction decomposition?

Step-by-Step Solution

Answer

Answer: Functions that can be integrated using partial fraction decomposition are rational functions with a factored denominator having a higher degree than the numerator's polynomial. These include rational functions with denominators factored into distinct linear factors or repeated linear factors.

Step 1: Understand Rational Functions

A rational function is a function that can be represented as a fraction of two polynomials (numerator and denominator). For a function to be called a rational function, both the numerator and denominator should be polynomials. An example of a rational function is R(x) = (x^2 + 4x + 3) / (x^2 - 1).

Step 2: Identify the Requirements for Partial Fraction Decomposition

Partial fraction decomposition is a technique used to simplify a given rational function into a sum or difference of simpler rational functions. This method is applicable when the denominator of the given rational function can be factored into distinct linear factors or repeated linear factors. In other words, partial fraction decomposition can be applied when the degree of the denominator's polynomial is greater than the degree of the numerator's polynomial, and the denominator can be factored.

Step 3: Recognize Functions to be Integrated using Partial Fraction Decomposition

Based on the requirements mentioned above, the following types of functions can be integrated using partial fraction decomposition: 1. Rational functions where the degree of the denominator's polynomial is greater than the degree of the numerator's polynomial and the denominator's polynomial can be factored into distinct linear factors. 2. Rational functions where the degree of the denominator's polynomial is greater than the degree of the numerator's polynomial and the denominator's polynomial can be factored into repeated linear factors. In summary, functions that can be integrated using partial fraction decomposition are rational functions with a factored denominator having a higher degree than the numerator's polynomial.