Integrating Rational Functions using Partial Fraction Decomposition
Introduction
- Partial Fraction Decomposition: A technique to simplify the integration of rational functions by breaking them down into simpler fractions.
- Key Requirement: Ensure the integral is completely factored.
Distinguishing Factors
- Linear Factors: Examples include
x, 3x, x + 4, 4x - 5.
- Quadratic Factors: Examples include
x^2, x^2 + 4, x^2 + 3x, x^2 + 2x + 7.
Steps to Partial Fraction Decomposition
- Factor the Denominator: Identify linear and quadratic factors.
- Break Down the Expression: Express as a sum of simpler fractions with constants on top.
- For linear factors: constants
A, B, etc.
- For quadratic factors: linear expressions like
Ax + B.
- Example: 1/4 + 1/5 becomes
9/20; partial fraction breaks it back to 1/4 + 1/5.
Solving for Constants
- Clear the Integral & Denominator: Set fraction equal to the sum of decomposed fractions.
- Use Shortcut: Plug specific
x values to eliminate terms and solve for constants A, B.
- Use values that zero out terms to simplify calculations.
Example Calculation
- Given: Integral of
1/(x^2 - 4).
- Linear Factors:
x + 2, x - 2.
- Calculate Constants:
- When
x = 2, solve for B.
- When
x = -2, solve for A.
Integration of Rational Functions
- Antiderivative of
1/x: ln |x|
- General Formula: ln of the linear expression divided by its derivative when integrating functions like
1/(ax + b).
Combining Logarithms
- Properties of Logs:
- ln(A) - ln(B) = ln(A/B)
- ln(A) + ln(B) = ln(A * B)
- Application: Use to simplify final expressions.*
Additional Practice
- Complex Denominators: Handle repeated factors, combine into single logarithmic expression.
- Quadratic Terms: Require linear numerators in decomposed fractions.
- Trig Substitution: Use for integrals involving squares that can relate to trigonometric identities.
Example Problems
- Repeated Linear Factors: Require multiple terms in decomposition (e.g.,
x - 2 squared needs A/(x-2) + B/(x-2)^2).
- Integration by Parts: Use when direct integration is complex.
- Trig Substitution for Quadratic Terms: Convert into secant/trigonometric identities.
Conclusion: Understanding partial fraction decomposition is crucial for integrating complex rational functions. Focus on factorization, identifying appropriate constants, and utilizing logarithmic properties to simplify results.