Integration of Rational Functions Using Partial Fraction Decomposition

Introduction

• Focus on integrating rational functions.
• Use of partial fraction decomposition.

Factoring

• Factor the denominator fully before decomposing.
• Example: Difference of perfect squares.
  • ( x^2 - 4 = (x + 2)(x - 2) ).
• Difference between linear and quadratic factors.
  • Linear Factors: ( x, 3x, x + 4 ).
  • Quadratic Factors: ( x^2, x^2 + 3x ).

Partial Fraction Decomposition

• Break a fraction into smaller fractions.
• For linear factors: ( \frac{A}{x+2} + \frac{B}{x-2} ).
• For quadratic factors: ( \frac{Ax+B}{x^2+8} ).

Example of Partial Fractions

• Simplifying ( \frac{9}{20} ) into ( \frac{1}{4} + \frac{1}{5} ).
• Process involves finding constants ( A, B, \ldots ).

Solving for Constants

• Multiply by the denominator to eliminate fractions.
• Use specific x-values to solve for constants.
  • Example: Solving for ( B ) by setting ( x = 2 ).
  • Solving for ( A ) by setting ( x = -2 ).

Integrating the Function

• Antiderivative of ( \frac{1}{x} ) is ( \ln |x| ).
• Example with constants: ( \frac{4}{x+7} ) becomes ( 4 \ln |x+7| ).

Properties of Logs

• Combine logs: ( \ln a - \ln b = \ln \frac{a}{b} ).
• Use in final integration expressions.

Practice Problems

• Practice with quadratic and repeated factors.
  • Example: ( \frac{x-4}{x^2 + 2x - 15} ).
  • Factor the denominator and find ( A, B ).

Handling Quadratic or Repeated Factors

• For quadratic: ( \frac{Ax+B}{x^2+4} ).
• For repeated: Use multiple terms: ( \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-2^2} ).

Complex Example

• Integrating ( \frac{x^2 + 9}{(x^2 - 1)(x^2 + 4)} ).
• Factoring and decomposing for integration.
• Use of ( \ln ) for linear terms and arctan for quadratic terms.

Trigonometric Substitution

• Substitute in complex quadratic cases.
• Example: ( x = 2\tan\theta ) leads to secant integration.

Final Result

• Combine all partial results for a complete integral expression.
• Use properties of logarithms to simplify results where possible.

Conclusion

• Integration using partial fraction decomposition requires careful factorization and solving for constants.
• Application of logarithmic properties aids in simplifying expressions.