Introduction
471
- 7.1Integration by Parts
- 7.2Integrals Containing Trigonometric Functions
- 7.3Integration Using Trigonometric Substitution: Integrands Containing \(\sqrt{a^{2}-x^{2}}\), \(\sqrt{x^{2}+a^{2}}\), or \(\sqrt{x^{2}-a^{2}}\), \(a>0\)
- 7.4Substitution: Integrands Containing \(ax^{2}+bx+c\)
- 7.5Integration of Rational Functions Using Partial Fractions
- 7.6Integration Using Numerical Techniques
- 7.7Integration Using Tables and Computer Algebra Systems
- 7.8Improper Integrals
- Chapter Review
- Chapter Project
The Birds of Rügen Island
Where else can a variety of shore and grassland birds, such as Redshanks, Lapwings, and Pied Avocet, find an undisturbed habitat? A department of the Western Pomeranian National Park administrates the Western Pomerania Lagoon Area and Jasmund National Park, in which Rügen Island lies. This area on the Baltic Sea off the coast of Germany includes some of the most important breeding grounds for water and mud flat birds of the Baltic Sea. How does a scientist model the population of birds that live in a protected area like Rügen Island?
CHAPTER 7 PROJECT
See the Chapter Project on page 535 for a discussion of how we can use calculus to model bird populations.
We have developed a sizable collection of basic integration formulas that are listed below. Some of the integrals are the result of simple antidifferentiation. Others, such as \(\int \tan x\,d\kern-.6ptx\), were found using the method of substitution, technique of integration. In this chapter, we further expand the list of basic integrals by exploring more techniques of integration.
Integration, unlike differentiation, has no hard and fast rules. It is often difficult, sometimes even impossible, to integrate a function that appears to be simple.
As you study the integration techniques presented in this chapter, it is important to recognize the form of the integrand for each technique. Then, with practice, integration becomes easier.
