**Unit 3: Techniques and Principles of Integration**
*Until now, we have been spending the majority of our time on the
integration of relatively simple functions (at least in comparison to
some of the functions we discussed in Part I). In this unit, we will
learn how to analyze more complex functions using more sophisticated
machinery. This includes clever methods of substitution, guides to
algebraic simplification, and integration by parts, as well as using
tables of integration or approximating the integral numerically. We
will also address how to manage integrals where either the integrand is
discontinuous in the domain of integration, or the domain of integration
is infinite.*

**Unit 3 Time Advisory**

This unit should take you 26.75 hours to complete.

☐ Subunit 3.1: 17.25 hours

☐ Sub-subunit 3.1.1: 3.25 hours

☐ Sub-subunit 3.1.2: 4 hours

☐ Sub-subunit 3.1.3: 2 hours

☐ Sub-subunit 3.1.4: 3.25 hours

☐ Sub-subunit 3.1.5: 4.75 hours

☐ Subunit 3.2: 1 hour

☐ Subunit 3.3: 2.75 hours

☐ Subunit 3.4: 5.75 hours

☐ Sub-subunit 3.4.1: 4.25 hours

☐ Sub-subunit 3.4.2: 1.5 hours

**Unit3 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:
- Use integration by parts to compute definite and
indefinite integrals.
- Integrate trigonometric functions.
- Use trigonometric substitution to compute definite and indefinite
integrals.
- Use the natural logarithm in substitutions to compute integrals.
- Integrate rational functions using the method of partial fractions.
- Compute integrals using integral tables.
- Approximate integrals using numerical integration techniques
including the trapezoidal rule and Simpson’s rule.
- Compute improper integrals of both types.

**3.1 Methods of Integration**
*So far, we have seen two ideas for computing integrals: directly apply
a formula (adjusting for coefficients and using the sum rule to break
apart polynomials and similar); or rewrite the integral in some way,
either by algebraic manipulation or by substitution. This subunit
expands the number of rewriting techniques at our disposal and adds a
new technique entirely: integration by parts.*

*Unlike differentiation, integration requires a little more forethought
or “creativity” in certain situations, as the correct implementation of
integration methods is less obvious. In fact, there are often multiple
correct methods to solve a complicated integral problem, though they may
vary in difficulty. It may not be clear whether the approach you are
using is correct until you are partway through the problem, so stick out
your attempt until you either succeed or hit an obvious wall.*

**3.1.1 Integration by Parts**
*Though this formula may at first seem arbitrary, integration by parts
is merely the product rule in reverse. With it, we use the information
we have to determine what the initial functions were. Integration by
parts is useful for integrands that are the product of two functions
from different “families,” such as an exponential with a polynomial.*

**Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.4: Integrals by Parts”**Link: University of Wisconsin: H. Jerome Keisler’s*Elementary Calculus*: Chapter 7: Trigonometric Functions: “Section 7.4: Integration by Parts” (PDF)Instructions: Please click on the link above, and read Section 7.4 in its entirety (pages 391 through 395). Integration by parts is a technique used to integrate more complicated combinations of functions. It is easy to derive – simply rearrange the product rule! Careful bookkeeping is essential for mastering this technique, so keep plenty of scrap paper on hand, use different variables if you have to perform integration by parts a second time in the same problem, and be neat. It will save you time in the end.

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

**Lecture: YouTube: MIT: David Jerison’s “Lecture 30: Integration by Parts, Reduction Formulae”**Link: YouTube: MIT: David Jerison’s “Lecture 30: Integration by Parts, Reduction Formulae” (YouTube)Also Available in:

iTunes UInstructions: Please watch the segment of Lecture 30 from time 18:20 minutes through the end. Note that lecture notes are available in PDF; the links are on the same pages as the lectures.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4)

**Assessment: University of California, Davis: Duane Kouba’s “The Method of Integration by Parts: Problems 1-23”**Link: University of California, Davis: Duane Kouba’s “The Method of Integration by Parts: Problems 1-23” (HTML)Instructions: Click on the link above and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.1.2 Trigonometric Integration**
*Trigonometric identities will be used heavily in this and the next
sub-subunit. If you wish to review trigonometric identities, they are
covered in the first three sections of Unit 3 of Precalculus II*
(MA003)*. Pay particular
attention to sin ^{2} x + cos^{2} x = 1 and its
counterparts for tan/sec and cot/csc, the half-angle formulas, and the
double-angle formulas.*

*Trigonometric integration is a simplification method: if you are asked
to integrate sin ^{8} x cos x, you can substitute for sin x and
be on your way. However, the situation is different if cos x also has a
larger exponent. This sub-subunit covers methods to “whittle down” the
exponents of such a problem until substitution applies.*

**Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.5: Integrals of Powers of Trigonometric Functions”**Link: University of Wisconsin: H. Jerome Keisler’s*Elementary Calculus*: Chapter 7: Trigonometric Functions: “Section 7.5: Integrals of Powers of Trigonometric Functions” (PDF)Instructions: Please click on the link above, and read Section 7.5 in its entirety (pages 397 through 401).

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

**Lecture: YouTube: MIT: Haynes Miller’s “Lecture 27: Trigonometric Integrals and Substitution” and “Lecture 28: Integration by Inverse Substitution; Completing the Square”**Link: YouTube: MIT: Haynes Miller’s “Lecture 27: Trigonometric Integrals and Substitution” (YouTube)Also Available in:

iTunes Uand “Lecture 28: Integration by Inverse Substitution; Completing the Square” (YouTube)

Also Available in:

iTunes UInstructions: Please click on the first link above, and watch the entirety of Lecture 27. Please watch Lecture 28 from the beginning up to time 15:50 minutes. Note that lecture notes are available in PDF; the links are on the same pages as the lectures. In these lectures, Dr. Miller will discuss how to integrate powers of trigonometric functions. At the end of Lecture 27, he will do an example related to trigonometric substitution, which will be the focus of the next section.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 15 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).

**Assessment: University of California, Davis: Duane Kouba’s “The Integration of Trigonometric Integrals: Problems 1-27”**Link: University of California, Davis: Duane Kouba’s “The Integration of Trigonometric Integrals: Problems 1-27” (HTML)Instructions: Click on the link above, and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours and 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.1.3 Trigonometric Substitution**
*Trigonometric substitution is a particular “inverse substitution”
technique. In substitution as we have most commonly seen so far, the
new variable is given as a function of the old variable; in inverse
substitution, this relationship is reversed. In trigonometric
substitution, we let our old variable be a trigonometric function of the
new variable, chosen so the Pythagorean Theorem applies to simplify the
original integrand. This technique is useful when you have a binomial
you wish were a monomial, typically because your binomial is on the
bottom of a fraction or under a radical.*

**Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.6: Trigonometric Substitutions”**Link: University of Wisconsin: H. Jerome Keisler’s*Elementary Calculus*: Chapter 7: Trigonometric Functions: “Section 7.6: Trigonometric Substitutions” (PDF)Instructions: Please click on the link above, and read Section 7.6 in its entirety (pages 402 through 405).

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

**Lecture: YouTube: MIT: Haynes Miller’s “Lecture 28: Integration by Inverse Substitution; Completing the Square”**Link: YouTube: MIT: Haynes Miller’s “Lecture 28: Integration by Inverse Substitution; Completing the Square” (YouTube)Also Available in:

iTunes UInstructions: Please watch Lecture 28 from time 15:50 minutes to the end. Note that lecture notes are available in PDF; the link is on the same page as the lecture.

Viewing this video and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).

**Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Methods of Integration”**Link: Temple University: Gerardo Mendoza’s and Dan Reich’s*Calculus on the Web: Calculus Book II*: Dan Reich’s “Methods of Integration” (HTML)Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “4. Methods of Integration,” and click button 164 (Trigonometric Substitution). Do problems 1-11. Note that the first problem has been done for you as an example; just click through the example in order to get a sense of the setup of the module. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.1.4 More Techniques Using the Natural Logarithm**
*If y = f(x), then ln y = ln (f(x)). If f(x) involves powers, products,
or quotients, taking the natural logarithm may allow us to simplify,
making differentiation easier. On the other hand, the fact that the
integral of du/u is ln u seems only narrowly applicable. However,
certain functions that do not originally appear to be of the form du/u
may be manipulated into that form. In particular, this approach allows
us to integrate powers of tangent and secant more easily.*

**Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.7: Derivatives and Integrals Involving ln(x)”**Link: University of Wisconsin: H. Jerome Keisler’s*Elementary Calculus*: Chapter 8: Exponential and Logarithmic Functions: “Section 8.7: Derivatives and Integrals Involving ln(x)” (PDF)Instructions: Please click on the link above, and read Section 8.7 in its entirety (pages 469 through 473).

Studying this reading should take approximately 30 minutes.

**Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function”**Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function” (QuickTime)Instructions: Please click on the link, scroll down to Video 31: “The Natural Logarithmic Function,” and watch from the 6th slide (marked 6 of 8) to the end. Feel free to watch the entire video if you would like a refresher on some earlier concepts. This short video gives examples of how to use the properties of the natural log to compute some more complicated integrals.

Studying this lecture should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Assessment: University of California, Davis: Duane Kouba’s “The Integration of Rational Functions, Resulting in Logarithmic of Arctangent Functions: Problems 1-22”**Link: University of California, Davis: Duane Kouba’s “The Integration of Rational Functions, Resulting in Logarithmic or Arctangent Functions: Problems 1-22” (HTML)Instructions: This assessment will cover sub-subunits 3.1.3 and 3.1.4. Click on the link above, and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours and 15 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.1.5 Rational Functions and Integration with Partial Fractions**
*You first learned about partial fractions in precalculus, when you
learned to re-write the quotient of two rational functions when the
denominator function can be written as the product of smaller factors.
Here, we use these methods to split apart a complicated fraction into
the sum of simpler fractions – in particular, fractions we know how to
integrate.*

*Polynomial factoring and long division will be used in this
sub-subunit. For a review of those topics, see Beginning Algebra*
(MA001)*, Unit 4 and subunit
3.4, respectively.*

**Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “4-10 Partial Fractions Expansions of Rational Functions”**Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s*Calculus and Linear Algebra Vol.1*: “4-10 Partial Fractions Expansions of Rational Functions” (HTML)Instructions: Please click on the link above, and read the assigned section. Use the “previous” and “next” links at the bottom of the webpage to navigate through this reading. This reading explains how to expand a rational function in terms of fractions of simpler rational functions. If you are not familiar with this method, this reading will be essential for understanding the rest of this section.

Studying this reading should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.8: Integration of Rational Functions”**Link: University of Wisconsin: H. Jerome Keisler’s*Elementary Calculus*: Chapter 8: Exponential and Logarithmic Functions: “Section 8.8: Integration of Rational Functions” (PDF)Instructions: Please click on the link above, and read Section 8.8 in its entirety (pages 474 through 480). Once you have mastered partial fractions, there are established methods for integrating rational functions. Careful, neat work will help you enormously in this section.

Studying this reading should take approximately 45 minutes.

**Lecture: YouTube: MIT: David Jerison’s “Lecture 29: Partial Fractions” and “Lecture 30: Integration by Parts, Reduction Formulae”**Link: YouTube: MIT: David Jerison’s “Lecture 29: Partial Fractions” (YouTube)Also Available in:

iTunes Uand “Lecture 30: Integration by Parts, Reduction Formulae” (YouTube)

Also Available in:

iTunes UInstructions: Please click on the first link above, and watch the entirety of Lecture 29. Please watch the segment of Lecture 30 from the beginning to time 18:20. Note that lecture notes are available in PDF; the links are on the same pages as the lectures.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 30 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).

**Assessment: University of California, Davis: Duane Kouba’s “The Method of Integration by Partial Fractions: Problems 1-20”**Link: University of California, Davis: Duane Kouba’s “The Method of Integration by Partial Fractions: Problems 1-20” (HTML)Instructions: Click on the link above, and work through all of the assigned problems. You will need to scroll down the page a bit to get to the problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.2 Integration with Tables & CAS (Computer Algebra Systems)**
*In theory, one could use the techniques we have learned so far to
integrate any integrable function. However, using tables allows us to
avoid “reinventing the wheel” and re-deriving the techniques for more
complicated but standard integrands by summarizing the results of the
process.*

**Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Integration Techniques: Using Integral Tables”**Link: Lamar University: Paul Dawkins’s*Paul’s Online Math Notes: Calculus II*: “Integration Techniques: Using Integral Tables” (HTML)Instructions: Please click the link above and read this entire section. In this section, Professor Dawkins gives some helpful hints about using integral tables to compute integrals quickly; this involves reducing whatever problem you are faced with to a problem in the tables

*.*He bases his discussion on the tables in the textbook used by his classes. However, it is easy to find tables of integrals on the Internet; we list several at the end of this page.Studying this reading should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.3 Numerical Integration**
*With numerical integration, we abandon the antiderivative and work with
modifications of the Riemann sum. This is possible to do for any
integrand, whereas antiderivatives do not exist for every integrand. In
most cases, we may also compute an error bound, allowing us to
approximate the definite integral to any degree of accuracy we like
(provided we have time to carry out all of the computations).*

**Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.6 “Numerical Integration”**Link: University of Wisconsin: H. Jerome Keisler’s*Elementary Calculus*: Chapter 4: Integration: “Section 4.6: Numerical Integration” (PDF)Instructions: Please click on the link above, and read Section 4.6 in its entirety (pages 224 through 233). In MA101, we used this text to introduce the trapezoidal rule. You will review that method for numerical integration and also learn about Simpson’s Rule in this reading. These approximation methods are used by mathematical software to calculate the values of definite integrals to very high degrees of accuracy.

Studying this reading should take approximately 1 hour.

**Lecture: YouTube: MIT: David Jerison’s “Lecture 24: Numerical Integration” and “Lecture 25: Exam Review”**Link: YouTube: MIT: David Jerison’s “Lecture 24: Numerical Integration” (YouTube)Also Available in:

iTunes Uand “Lecture 25: Exam Review” (YouTube)

Also Available in:

iTunes UInstructions: Please click on the first link above, and watch the segment of Lecture 24 from time 33:50 minutes through the end. Then, click on the second link above, and watch Lecture 25 from the beginning up to time 14:04 minutes. Note that lecture notes are available in PDF; the links are on the same pages as the lectures. In these lectures, Dr. Jerison will discuss motivation and methods for numerical integration, including the trapezoidal rule and Simpson’s rule.

Viewing these lectures and pausing to take notes should take approximately 45 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).

**Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Numerical Integration”**Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Numerical Integration” (PDF)Also Available in:

Instructions: Please click on the link above, and work through each of the six examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

**3.4 Improper Integrals**
*In previous integration examples, we have either ignored the domain of
the function (i.e. antiderivatives) or integrated over an interval
without any discontinuities. But what if we want to integrate over an
interval tending toward infinity, or what if we want to find the area
under a curve on an interval with a vertical asymptote? This subunit
will introduce you to these integrals, which we refer to as “improper
integrals.”*

**3.4.1 Type I**
- **Reading: Lamar University: Paul Dawkins’s Paul’s Online Math
Notes: Calculus II: “Integration Techniques: Improper Integrals”**
Link: Lamar University: Paul Dawkin’s *Paul’s Online Math Notes:
Calculus II*: “Integration Techniques: Improper
Integrals” (HTML)

```
Instructions: This reading will cover sub-subunits 3.4.1 and
3.4.2. Please click the link above, and read this entire section.
The two types of improper integrals are Type I, those with
“infinite” limits of integration, and Type II, those where the
integrand has infinite discontinuities somewhere in the interval of
integration. Professor Dawkins does not use this terminology,
although it is common, but he does discuss how to deal with each
type of improper integral.
Studying this reading should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**Lecture: YouTube: MIT: David Jerison’s “Lecture 36: Improper Integrals” and “Lecture 37: Infinite Series and Convergence Tests”**YouTube: MIT: David Jerison’s “Lecture 36: Improper Integrals” (YouTube)Also Available in:

iTunes Uand “Lecture 37: Infinite Series and Convergence Tests” (YouTube)

Also Available in:

iTunes UInstructions: These lectures will cover sub-subunits 3.4.1 and 3.4.2. Please click on the first link above, and watch Lecture 36 from time 3:22 minutes to the end. Please click on the second link above, and watch Lecture 37 from the beginning up to time 17:35 minutes. Note that lecture notes are available in PDF; the link is on the same page as the lecture. In these lectures, Professor Jerison will discuss how to estimate as well as evaluate improper integrals of both types.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 15 minutes.

**Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Improper Integrals over Unbounded Intervals”**Link: Temple University: Gerardo Mendoza’s and Dan Reich’s*Calculus on the Web: Calculus Book II*: Matthias Beck and Molly M. Cow’s “Improper Integrals over Unbounded Intervals” (HTML)Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “6. Improper Integrals,” and click button 172 (Improper Integrals over Unbounded Intervals). Do problems 1-18. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 2 hours.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.4.2 Type II**
- **Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s
Calculus on the Web: Calculus Book II: Matthias Beck and Molly M.
Cow’s “Improper Integrals of Unbounded Functions”**
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s *Calculus
on the Web: Calculus Book II*: Matthias Beck and Molly M. Cow’s
“Improper Integrals of Unbounded
Functions” (HTML)

```
Instructions: Click on the link above. Then, click on the “Index”
button. Scroll down to “6. Improper Integrals,” and click button
173 (Improper Integrals of Unbounded Functions). Do all 16
problems. If at any time a problem set seems too easy for you, feel
free to move on.
Completing this assessment should take approximately 1 hour and 30
minutes.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```