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3: Techniques of Integration

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Apr 1, 2025
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- 2.10: Chapter 2 Review Exercises
- 3.1: Integration by Parts

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We saw in the previous chapter how important integration can be for all kinds of different topics—from calculations of volumes to flow rates, and from using a velocity function to determine a position to locating centers of mass. It is no surprise, then, that techniques for finding antiderivatives (or indefinite integrals) are important to know for everyone who uses them. We have already discussed some basic integration formulas and the method of integration by substitution. In this chapter, we study some additional techniques, including some ways of approximating definite integrals when normal techniques do not work.

In a large city, accidents occurred at an average rate of one every three months at a particularly busy intersection. After residents complained, changes were made to the traffic lights at the intersection. It has now been eight months since the changes were made and there have been no accidents. Were the changes effective or is the eight-month interval without an accident a result of chance? We explore this question later in this chapter and see that integration is an essential part of determining the answer.

This is a picture of a city street with a traffic signal. The picture has very busy lanes of traffic in both directions. — Figure 3.1.1: Careful planning of traffic signals can prevent or reduce the number of accidents at busy intersections. (Creative Commons Attribution-Share Alike 3.0; David McKelvey via Flickr)

3.1: Integration by Parts
This page provides an overview of integration by parts, a technique used to simplify the integration of products of functions. It includes the formula derived from the product rule, guidance on choosing functions with the LIATE mnemonic, and multiple examples ranging from logarithmic to trigonometric integrals. The text also discusses evaluating both definite and indefinite integrals and emphasizes validating results through differentiation.
- 3.1E: Exercises for Section 3.1
3.2: Trigonometric Integrals
Trigonometric substitution is an integration technique that allows us to convert algebraic expressions that we may not be able to integrate into expressions involving trigonometric functions, which we may be able to integrate using the techniques described in this section. In addition, these types of integrals appear frequently when we study polar, cylindrical, and spherical coordinate systems later. Let’s begin our study with products of sin x and cos x.
- 3.2E: Exercises for Section 3.2
3.3: Trigonometric Substitution
The technique of trigonometric substitution comes in very handy when evaluating integrals of certain forms. This technique uses substitution to rewrite these integrals as trigonometric integrals.
- 3.3E: Exercises for Section 3.3
3.4: Partial Fractions
In this section, we examine the method of partial fraction decomposition, which allows us to decompose rational functions into sums of simpler, more easily integrated rational functions.
- 3.4E: Exercises for Section 3.4
3.5: Other Strategies for Integration
In addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which are readily available in many books, including the appendices to this one. Also widely available are computer algebra systems (CAS), which are found on calculators and in many campus computer labs, and are free online.
- 3.5E: Exercises for Section 3.5
3.6: Numerical Integration
The antiderivatives of many functions either cannot be expressed or cannot be expressed easily in closed form (that is, in terms of known functions). Consequently, rather than evaluate definite integrals of these functions directly, we resort to various techniques of numerical integration to approximate their values. In this section we explore several of these techniques. In addition, we examine the process of estimating the error in using these techniques.
- 3.6E: Exercises for Section 3.6
3.7: Improper Integrals
In this section, we define integrals over an infinite interval as well as integrals of functions containing a discontinuity on the interval. Integrals of these types are called improper integrals. We examine several techniques for evaluating improper integrals, all of which involve taking limits.
- 3.7E: Exercises for Section 3.7
3.8: Probability
A variable, say 𝑋XX, that can take certain values, each with a corresponding probability, is called a random variable; in the example above, the random variable was the sum of the two dice. If the possible values for X are $𝑥_1,𝑥_2,𝑥_3,.......𝑥_𝑛$ then the expected value of the random variable is 𝐸(𝑋)=∑𝑛𝑖=1𝑥𝑖𝑃(𝑥𝑖)E(X)=∑i=1nxiP(xi) E(X)=\sum_{i=1}^n x_iP(x_i). The expected value is also called the mean.
- 3.8E: Exercises for section 3.8
3.9: Chapter 3 Review Exercises
This page reviews integral calculus concepts through exercises on integration methods, numerical techniques for approximating integrals, and improper integrals. It covers integration by parts, partial fractions, and trigonometric substitution, as well as evaluating convergence and the gamma function. The use of technology for analysis is illustrated, including the velocity of a Bugatti Veyron, alongside specific answers for the exercises.

Thumbnail: Using Riemann Sum to approximate the area under the curve. (CC BY NC SA; Openstax via Calculus-volume-2)

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