ITLR - Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions

Partial Fraction Decomposition

$Did You Know? The method of partial fractions was introduced by John Bernoulli (1667-1748), a Swiss mathematician who was instrumental in the early development of calculus. His most famous student was Leonard Euler.$ In preparation for evaluating antiderivatives expressed as rational functions, a review of the technique of partial fraction decomposition may be helpful. Partial fraction decomposition is a method of transforming a rational function (a ratio of polynomials) into a sum of simpler fractions. Its use is dependent on the factorability of the denominator. The presentation below outlines the rationale and requisite conditions for partial fraction decomposition and illustrates how this technique is applied to rational functions involving linear factors.

Watch the video on Partial Fraction Decomposition Part 1 Links to an external site.

When rational functions involve quadratic factors of the denominator, a modification of the technique applied for linear factors is necessary. The next presentation focuses on partial fraction decomposition involving quadratic factors.

Watch the video on Partial Fraction Decomposition Part 2 Links to an external site.

In general, steps to decompose a rational function into partial fractions include:

1. Divide if the rational function is an improper fraction, i.e., if the degree of the numerator > the degree of the denominator. Express the result as a polynomial plus a proper fraction.

2. Factor the denominator of the proper fraction into irreducible linear (px + q)^m and/or quadratic (ax² + bx + c)ⁿ factors.

3. For each linear factor of the form (px + q)^m, where m is a positive integer and p, q, and A_i are real constants, include the following sum of m fractions in the partial fraction decomposition.

equation image indicator $LaTeX: \frac{A_1}{px+q}+\frac{A_2}{\left(px+q\right)^2}+...+\frac{A_m}{\left(px+q\right)^m}$ $\frac{A_1}{px+q}+\frac{A_2}{\left(px+q\right)^2}+...+\frac{A_m}{\left(px+q\right)^m}$

4. For each quadratic factor of the form (ax² + bx + c)ⁿ, where n is a positive integer and a, b, c, B_i and C_i are real constants, include the following sum of n fractions in the partial fraction decomposition.

equation image indicator $LaTeX: \frac{B_1x+C_1}{ax^2+bx+c}+\frac{B_2x+C_2}{\left(ax^2+bx+c\right)^2}+...+\frac{B_nx+C_n}{\left(ax^2+bx+c\right)^n}$ $\frac{B_1x+C_1}{ax^2+bx+c}+\frac{B_2x+C_2}{\left(ax^2+bx+c\right)^2}+...+\frac{B_nx+C_n}{\left(ax^2+bx+c\right)^n}$

5. Set the original proper fraction equal to the sum of partial fractions and clear the resulting equation of fractions.

6. Equate the coefficients of corresponding powers of x and solve the resulting equations for the undetermined coefficients.

Integration of Rational Functions with Linear Factors

A method for finding antiderivatives of rational functions that depends on partial fraction decomposition is integration by partial fractions. Integral evaluation involving denominators consisting of linear factors and utilizing the method of partial fractions is illustrated in the next presentation.

Watch the video on Integration Partial Fraction Decomposition Part 1 Links to an external site.

The following presentation illustrates the method of partial fractions applied to integrating rational functions with non-repeated and repeated linear factors.

$method of partial fractions one image$ $method of partial fractions two image$

Integration of Rational Functions with Quadratic Factors

The technique for decomposing partial fractions involving denominators with quadratic factors is now applied to finding antiderivatives. View the presentation that illustrates integration using partial fractions with linear and quadratic factors.

Watch the video on Integration Partial Fraction Decomposition Part 2 Links to an external site.

Indeterminate Forms and L'Hopital's Rule Practice

Indeterminate Forms and L'Hopital's Rule: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of integration of rational functions by partial fractions.

IMAGES CREATED BY GAVS