Question

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

Short answer

Answer: Partial fraction decomposition can be used to integrate rational functions meeting the following criteria: 1. Proper rational functions, where the degree of the numerator is strictly less than the degree of the denominator, and the denominator can be factored into linear or quadratic factors. 2. Improper rational functions, after performing polynomial division to write the function as a sum of a polynomial and a proper rational function. 3. Proper rational functions with repeated linear or quadratic factors in the denominator. The conditions that must be met are: 1. The degree of the numerator must be strictly less than the degree of the denominator. 2. The denominator must be factorable into linear or quadratic factors (or a combination of both) with real or complex coefficients.

Step-by-step solution

Rational Functions

A rational function is a function that can be represented as a quotient of two polynomials, i.e., R(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

Requirements for Partial Fraction Decomposition

Partial fraction decomposition is a method used to integrate rational functions. To use this method with a rational function R(x) = P(x)/Q(x), two conditions must be met: 1. The degree of the numerator P(x) must be strictly less than the degree of the denominator Q(x). If the degree of P(x) is greater than or equal to the degree of Q(x), we must first perform polynomial division to get a proper rational function. 2. The denominator Q(x) must be factorable into linear or quadratic factors (or a combination of both) with real or complex coefficients. If Q(x) is an irreducible polynomial, then partial fraction decomposition can't be applied directly.

Types of Rational Functions that can be Integrated using Partial Fraction Decomposition

Based on the requirements mentioned above, the following types of rational functions can be integrated using partial fraction decomposition: 1. Proper rational functions, where the degree of the numerator is strictly less than the degree of the denominator, and the denominator can be factored into linear or quadratic factors. Example: \[ \frac{x}{(x^2+1)(x-2)} \] 2. Improper rational functions, where the degree of the numerator is greater than or equal to the degree of the denominator, after performing polynomial division to write the function as a sum of a polynomial and a proper rational function. Example: \[ \frac{x^3 - 2x^2 + 3x}{x^2 - 4} = (x - 3) + \frac{6x - 12}{x^2 - 4} \] 3. Proper rational functions with repeated linear or quadratic factors in the denominator. Example: \[ \frac{x^2 + 1}{(x^2 - 4)^2} \] In conclusion, partial fraction decomposition can be used to integrate rational functions that meet the above-mentioned criteria. By identifying the form of the given function and following the steps of the partial fraction decomposition, one can successfully integrate the function.