Integrating using linear partial fractions

Integrating using linear partial fractions is a method used to simplify the integration of rational functions, which are expressed as the ratio of two polynomials. The main ideas include:

Form of the Rational Function: Ensure the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first.
Factor the Denominator: Factor the denominator into linear factors (e.g., $(x - a)(x - b)$ ) or irreducible quadratic factors.
Set Up Partial Fractions: Write the rational function as a sum of simpler fractions. For linear factors, each term will be in the form $\frac{A}{x - a}$ for each linear factor and for repeated factors, it will include terms like $\frac{A}{(x - a)^n}$ .
Solve for Coefficients: Multiply both sides by the denominator to eliminate fractions, then solve for the unknown coefficients by substituting suitable values or equating coefficients.
Integrate Each Term: Once expressed in partial fractions, integrate each term separately. Linear terms typically integrate to logarithmic forms, while constants integrate to linear terms.
Combine Results: Add the results of the individual integrals to get the final answer, including the constant of integration.

Using this method simplifies complex integrals, making them easier to evaluate.

In this article

Part 1: Integration with partial fractions

Finding the integral of a rational function using linear partial fraction decomposition.

Sure! Here are the key points to learn when studying "Integration with Partial Fractions":

Understanding Rational Functions: Recognize that partial fraction decomposition is used for integrating rational functions (the ratio of two polynomials).
Proper vs. Improper Fractions: Ensure the rational function is a proper fraction (the degree of the numerator is less than the degree of the denominator). If it's improper, perform polynomial long division first.
Factoring the Denominator: Factor the denominator completely into linear and/or irreducible quadratic factors.
Setting Up Partial Fractions: Express the rational function as a sum of simpler fractions:
- For linear factors: $\frac{A}{(x-a)}$
- For irreducible quadratic factors: $\frac{Bx + C}{(x^2 + bx + c)}$
Solving for Coefficients: Multiply both sides by the common denominator to eliminate fractions, then solve for unknown coefficients (A, B, C, etc.) by substituting convenient values of x or by equating coefficients.
Integration: Integrate each term of the decomposed fraction separately. Use basic integration techniques:
- For $\frac{A}{(x-a)}$ : $\int \frac{A}{(x-a)} \,dx = A \ln |x - a|$
- For $\frac{Bx + C}{(x^2 + bx + c)}$ : Use substitution or other methods as needed.
Combining Results: Combine the results of the integrals and add the constant of integration $C$ .

By mastering these points, you can effectively apply partial fraction decomposition to integrate rational functions.

Integrating using linear partial fractions

Part 1: Integration with partial fractions

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