Math 192   Calculus 2         aacchead.jpg (10305 bytes)

Brief Study Guide for Chapter 8

Preview For Chapter 8

Techniques of Integration
In Calculus 1, integration was limited to a single substitution and reversing a derivative formula. In Chapter 8 we integrate far more complex integrands involving multiple substitutions.
We integrate algebraically with four techniques:
Integration by parts,
Trigonometric integrals,
Trigonometric substitution, and
Partial fractions decomposition.
We integrate using integral tables, found inside the back cover of Stewart.
We integrate using a computer algebra system, namely, Maple.
We integrate numerically, adding Simpson's rule to the list of numerical methods which include the trapezoidal and midpoint rules from Chapter 4.
Finally we integrate integrals involving infinity, called improper integrals.

Calculus 2     Lesson 8.1     Integration by Parts
The following topic is the most important. Typical exercises from Calculus, 5th edition, by James Stewart, are assigned at the end of the objective.
1. Integrate by parts.   P516#3,5,7,13,25,33(Let z=x1/2.),51
Maple Assignment:  See Maple Assignment

Calculus 2     Lesson 8.2     Trigonometric Integrals
1. Integrate trigonometric integrals. P524#1,5,9,25,29,55
  P520 The basic strategy for integrating sin(x) to a power, cos(x) to a power, or the product of both sin and cos each to some power is given in the box on the page.  Look at examples 1, 2, 3, and 4.
  P522 The basic formula for the integral of tan(x) is given above box 1and sec(x) is given in box 1.  
  P522 The basic strategy for integrating tan(x) to a power, sec(x) to a power, or the product of both tan and sec each to a power is given in the box on this page.  Look at examples 5, 6 7, and 8.
    There are several "poems" to help us remember the combinations of trigonometric integrals:

      SEcant Even Easy!

      Sine, Cosine or Tangent Odd, Not Hard!   (Do they call this poetic license?)

      Tangent Even, Secant Odd, Very, Very Hard!!!

These poems are illustrated in the exercises above as well as the Maple Session below.
Maple Assignment:  See Maple Assignment

Calculus 2     Lesson 8.3     Trigonometric Substitution
1. Integrate using trigonometric substitution. P530#1,3,5,7,11,15,23
Maple Assignment:  See Maple Assignment

Calculus 2     Lesson 8.4     Partial Fractions Decomposition
1. Integrate rational functions using partial fractions decomposition. P540#1-17 odd, 39, 45, 47
Maple Assignment:  See Maple Assignment

Calculus 2     Lesson 8.5    Strategy for Integration
1. Integrate. P546#1-35 odd
Maple Assignment:  none 

Calculus 2     Lesson 8.6     Integration Tables 
and Computer Algebra Systems
1. Integrate using integral tables. P551#1,5,9,35,37
2. Integrate using a computer algebra system (Maple).
Graphing Calculator Hints show how to evaluate derivatives and definite integrals using the TI-86.
Maple Assignment:  none

Calculus 2     Lesson 8.7     Approximate Integration
1.

Use a graph, table or formula of a function with the Midpoint rule, Trapeziodal rule and Simpson's rule to find the approximate value of its definite integral (Mn, Tn, and Sn) P563#7,33,35
2. Estimate the error EM, ET, and ES for a given definite integral and value of n. P563#19,25
Maple Assignment:  See Maple Assignment

Calculus 2     Lesson 8.8     Improper Integrals
1.

Determine whether improper integrals having infinite limits of integration converge and if so to what value. P573#5,7,9,13
2. Determine whether improper integrals with discontinuous integrands converge and if so to what value. P573#27,29,37,41
Maple Assignment:  See Maple Assignment

Maple worksheets to be submitted for a grade for Chapter 8: 

See Maple Assignment