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Calculus and Analytic Geometry II

Calculus and Analytic Geometry II. Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin. Chapter 08: Techniques of Integration. 08.01 Integration by Parts 08.02 Trigonometric Integrals 08.03 Trigonometric Substitutions

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Calculus and Analytic Geometry II

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  1. Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin

  2. Chapter 08: Techniques of Integration • 08.01 Integration by Parts • 08.02 Trigonometric Integrals • 08.03 Trigonometric Substitutions • 08.04 Integration of Rational Functions by Partial Fractions • 08.05 Integral Tables and Computer Algebra Systems • 08.06 Numerical Integration • 08.07 Improper Integrals

  3. Chapter 08 Overview • Techniques beyond memorization, guessing, and substitution are developed for finding indefinite integrals of more complicated integrands. • Computer Algebra Systems (CAS) are used extensively by mathematicians, physicists, engineers and practitioners of other technical disciplines for finding the indefinite integral of nearly any particular function.

  4. 08.01: Integration by Parts 1 • The Integration by Parts formula. Examples 1 & 6 • The integral of the natural logarithm function. Example 2 • Repeated use of Integration by Parts. Example 3 • Solving for an integral. Example 4 • Reduction formulas.

  5. 08.02: Trigonometric Integrals 1 • Products of Powers of Sines and Cosines (three cases, zero is considered to be an even number). Examples 1 – 3 • Eliminating Square Roots. Example 4 • Powers of tan [x] and sec [x]. Examples 5 & 6 • Products of Sines and Cosines. Example 7

  6. 08.03: Trigonometric Substitutions 1 • Trigonometric substitutions are used to change the variable in an integral to the angle in a related triangle. Examples 1 – 3

  7. 08.04: Integration of Rational Functions by Partial Fractions 1 • Obtaining partial fractions by Equating Coefficients. • Distinct linear factors. Example 1 • Repeated linear factors. Example 2 • Improper fractions. Example 3 • Irreducible quadratic factors. Example 4 • Repeated irreducible quadratic factors Example 5 • Obtaining partial fractions by using Differentiation and Identity Substitution. Example 8 • Obtaining partial fractions by Identity Substitution. Example 9 • Finding partial fractions with Wolfram Alpha (Apart).

  8. 08.05: Integral Tables and Computer Algebra Systems 1 • Using Wolfram Alpha to evaluate Nonelementary Integrals. Nonelementary Integrals

  9. 08.06: Numerical Integration 1 • This section is not covered.

  10. 08.07: Improper Integrals 1 • Type I Integrals: Infinite limits. Examples 1 & 2 • Integrating the reciprocal power function. Example 3 • Type II Integrals: Infinite discontinuities within the integral interval. Examples 4 & 5 • Evaluating Improper Integrals with Wolfram Alpha. • Table: Types of Improper Integrals Discussed in This Section.

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