Overview
This lecture covers how to integrate rational functions using the technique of partial fraction decomposition, including cases with distinct linear, repeated, and quadratic factors.
Factoring and Identifying Factors
- Always factor the denominator completely before starting partial fraction decomposition.
- Linear factors look like x, x + a, or kx + b (no x² term).
- Quadratic factors include x² or expressions like x² + bx + c.
Setting Up Partial Fractions
- For distinct linear factors: set up fractions like A/(x + a), B/(x + b).
- For irreducible quadratic factors: numerator is linear, e.g., (Cx + D)/(quadratic factor).
- For repeated linear factors: include terms for each power up to the exponent, e.g., A/(x − a) + B/(x − a)², etc.
Solving for Coefficients
- Multiply both sides by the denominator to clear fractions.
- Plug in x values that zero out terms to solve for unknowns.
- If more variables remain, use convenient x values to create a system and solve.
Integrating the Partial Fractions
- ∫1/(x + a) dx = ln|x + a| + C.
- ∫k/(x + a) dx = k·ln|x + a| + C.
- ∫1/(ax + b) dx = (1/a)·ln|ax + b| + C.
- For ∫1/(x² + a²) dx, answer is (1/a)·arctan(x/a) + C.
Working Through Examples
- Combine log expressions: ln A − ln B = ln(A/B).
- Coefficients can be moved to exponents: k·ln x = ln xᵏ.
- For repeated factors, use substitution if needed (e.g., ∫1/(x − a)² dx = −1/(x − a) + C).
- Use trigonometric substitution or the arctan formula for irreducible quadratics.
Key Terms & Definitions
- Rational function — a fraction of two polynomials.
- Partial fraction decomposition — expressing a rational function as a sum of simpler fractions.
- Linear factor — a factor of the form (x + a) or (ax + b).
- Quadratic factor — a factor involving x², such as (x² + a).
- Irreducible quadratic — quadratic that can't be factored over the reals.
- Antiderivative — the integral or primitive function.
- Arc tangent (arctan) — inverse tangent function, used in integrals of the form 1/(x² + a²).
- Constant of integration (C) — arbitrary constant added after integration.
Action Items / Next Steps
- Practice factoring denominators completely before setting up partial fractions.
- Solve the assigned example integrals using partial fraction decomposition.
- Review integration rules for logarithmic and arctan forms.
- Prepare for problems involving repeated and quadratic factors on upcoming assignments.