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

Chapter 08: Techniques of Integration Integration by Parts Trigonometric Integrals Trigonometric Substitutions Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems Numerical Integration Improper Integrals

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.

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.

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

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

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).

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

08.06: Numerical Integration 1 This section is not covered.

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.