Overview
This lecture reviews key integration techniques for AP Calculus BC, focusing on partial fractions, improper integrals, and integration by parts, with practice examples and guidance on notation and exam strategies.
Integration Techniques Overview
- AP Calculus BC requires mastering integration by substitution, partial fractions, improper integrals, and integration by parts.
- Identify the integration technique by analyzing the integrand's structure, such as factorizable denominators or products.
- Use proper mathematical notation and show steps clearly for full credit on FRQs.
Partial Fractions
- Use partial fractions when integrating a rational function with a factorizable denominator.
- Decompose the fraction into sums of simpler fractions with unknown constants (e.g., A/(x-a) + B/(x-b)).
- Solve for constants using algebraic or heavy side (cover-up) method by plugging in convenient x-values.
- Only linear, non-repeating denominators will be tested on the AP exam.
Improper Integrals
- An improper integral has an infinite limit or a discontinuity (e.g., vertical asymptote) in the interval.
- Replace infinity or the point of discontinuity in limits with a variable, then take the limit as that variable approaches the critical value.
- Split integrals at discontinuities and use proper limit notation to earn all points.
- If the limit does not exist (e.g., results in ln(0)), the integral diverges.
Integration by Parts
- Use integration by parts (∫u dv = uv - ∫v du) when integrating products of functions.
- Choose u and dv using mnemonics like LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential).
- Write u and dv diagonally; derive du (downward) and integrate dv (upward) for v.
- Only one iteration of integration by parts is typically needed on the exam.
Problem-Solving Tips & Multiple Choice Strategies
- Factor denominators to identify partial fractions or other applicable methods.
- When using natural logarithms, remember log(a) - log(b) = log(a/b).
- For definite integrals with vertical asymptotes within the interval, split and analyze each part as an improper integral.
- Carefully match your work to given answer choices by simplifying or combining log expressions as needed.
Free Response Problem Approach
- Recognize inverse trigonometric integration forms, e.g., ∫dx/(x²+a²) leads to arctan(x/a).
- For ∫dx/(x² - a²), factor and use partial fractions.
- Always plug in upper and lower bounds after integrating and simplify as needed.
- For improper integrals or limits to infinity, properly set up and evaluate using limit processes.
Key Terms & Definitions
- Partial Fractions — Decomposition of a rational expression into simpler fractions for easier integration.
- Improper Integral — An integral with infinite limits or a discontinuity inside the limits.
- Integration by Parts — Technique for integrating products, following ∫u dv = uv - ∫v du.
- Heavy Side/Cover-Up Method — Shortcut for finding constants in partial fraction decomposition.
- LIATE — Mnemonic to help choose u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
Action Items / Next Steps
- Download and complete extra integration practice problems from the provided link/QR code.
- Review and memorize basic integration formulas and the integration by parts formula.
- Practice setting up and notating improper integrals with correct limit notation.
- Watch for upcoming differential equations review focusing on logistic growth.