Chapter 8 Techniques of Integration

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, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Integrate by parts. P493#3,5, 7,15,27,33(Let z=x^1/2.),53
	P489	The basic formula for integration by parts is found in boxes 1 and 2.
	P489	Examples 1, 2 and 5 illustrate the basic formula.
	P491	Examples 3 and 4 illustrate cases where the formula has to be used more than once.
	P493	Example 6 illustrates the use of this technique in deriving a reduction formula.
Maple Assignment: See Maple Assignment Submitting Maple Assignments for a Grade

Calculus 2 Lesson 8.2 Trigonometric Integrals
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Integrate trigonometric integrals. P501#1,5,9,25,29,57
	P498	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.
	P500	The basic formula for the integral of tan(x) is given above box 1 and sec(x) is given in box 1.
	P499	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
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Integrate using trigonometric substitution. P508#1,3,5,7,11,15,23
	P503	The table for the trigonometric substitutions is found on this page.
	P504	The a² - u² type is found in examples 1, 2 and 7.
	P505	The a² + u² type is found in examples 3, 4 and 6.
	P506	The u² - a² type is found in example 5.
Maple Assignment: See Maple Assignment

Calculus 2 Lesson 8.4 Partial Fractions Decomposition
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Integrate rational functions using partial fractions decomposition. P517#1-17 odd, 39, 45, 47
	P510	Example 1 shows the long division process to change the integrand from a rational function to a a sum.
	P511	Examples 2 and 3 show the process when the rational function has a denominator with distinct linear factors.
	P513	Example 4 shows the process when the rational function has a denominator with repeated linear factors.
	P514	Examples 5 and 6 show the process when the rational function has a denominator with distinct irreducible quadratic factors.
	P516	Examples 7 and 8 show the process when the rational function has a denominator with repeated irreducible quadratic factors.
	P517	Example 9 shows the process when the rational function has both radical and polynomial parts.
Maple Assignment: See Maple Assignment

Calculus 2 Lesson 8.5 Strategy for Integration
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Integrate. P524#1-35 odd
	P520	A table giving basic integration formulas is found on this page. You should have these formulas memorized already.
	P520	The basic steps in the strategy are given on this page and the next.
	P522	Examples 1 through 5 show how the steps are used.
Maple Assignment: none

Calculus 2 Lesson 8.6 Integration Tables and Computer Algebra Systems
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Integrate using integral tables. P529#1,5,9,37,39
	P525	Tables of Integrals are found in the back of the text. Look carefully at the form of the problem and match the form of the problem to the particular integral in the table. Use substitutions to simplify the form. See examples 1, 2, 3, and 4.
2.		Integrate using a computer algebra system (Maple).
	P527	The basic method is to enter the function and have Maple integrate it with the int command. Remember to add the constant of integration , if necessary. Look at examples 5, 6, and 7.
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
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

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 (M_n, T_n, and S_n) P541#7,33,35
	P532	The Midpoint rule formula is given at the bottom of this page. Look at Example 1.
	P533	The Trapezoidal rule formula is given at the top of this page. Look at Examples 1 and 3.
	P538	The Simpson's rule formula is given at the top of this page. Look over the derivation on the previous page. Look at Examples 4, 5 and 7.
2.		Estimate the error E_M, E_T, and E_S for a given definite integral and value of n. P541#19,25
	P535	The error formula for the Midpoint rule is given in box 3. Use algebraic or graphical methods to get a bound for the second derivative. Look at Example 2.
	P535	The error formula for the Trapezoidal rule is given in box 3. Use algebraic or graphical methods to get a bound for the second derivative. Look at Examples 2 and 3.
	P539	The error formula for the Simpson's rule is given in box 4. Use algebraic or graphical methods to get a bound for the fourth derivative. Look at Examples 6 and 7.
Maple Assignment: See Maple Assignment

Calculus 2 Lesson 8.8 Improper Integrals
The following topic is the most important. Typical exercises from Calculus, 6^th edition, by James Stewart, are assigned at the end of the objective. In smaller print following the objective is a guide for the reading assignment in the form of page numbers in the text and notes regarding the reading about the objective.

1.		Determine whether improper integrals having infinite limits of integration converge and if so to what value. P551#5,7,9,13
	P545	The definition of the improper integral with infinite limits is given in box 1. Memorize it. Look at examples 1, 2, and 3.
	P547	Box 2 is an important consequence of this definition. We will need it in Chapter 12 when we look at series. See example 4.
2.		Determine whether improper integrals with discontinuous integrands converge and if so to what value. P551#27,29,35,41
	P548	The definition of the improper integral with the function discontinuous at an endpoint (upper or lower limit) is found in box 3. Look at examples 5, 6, and 8.
	P548	The definition of the improper integral with the function discontinuous between the endpoints (upper and lower limit) is found in box 3. Look at example 7.
Maple Assignment: See Maple Assignment