- 6.1: Substitution
- This chapter is devoted to exploring techniques of antidifferentiation. While not every function has an antiderivative in terms of elementary functions (a concept introduced in the section on Numerical Integration), we can still find antiderivatives of a wide variety of functions.
- 6.2: Integration by Parts
- Integration by parts is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be derived in one line simply by integrating the product rule of differentiation.
- 6.3: Trigonometric Integrals
- Functions involving trigonometric functions are useful as they are good at describing periodic behavior. This section describes several techniques for finding antiderivatives of certain combinations of trigonometric functions.
- 6.4: Trigonometric Substitution
- We have since learned a number of integration techniques, including Substitution and Integration by Parts, yet we are still unable to evaluate the above integral without resorting to a geometric interpretation. This section introduces Trigonometric Substitution, a method of integration that fills this gap in our integration skill. This technique works on the same principle as "normal" Substitution discussed previously.
- 6.5: Partial Fraction Decomposition
- In this section we investigate the antiderivatives of rational functions. It can be shown that any polynomial can be factored into a product of linear and irreducible quadratic terms. The key idea states how to decompose a rational function into a sum of rational functions whose denominators are all polynomials of lower degree.
- 6.6: Hyperbolic Functions
- The hyperbolic functions are a set of functions that have many applications to mathematics, physics, and engineering. Among many other applications, they are used to describe the formation of satellite rings around planets, to describe the shape of a rope hanging from two points, and have application to the theory of special relativity. This section defines the hyperbolic functions and describes many of their properties, especially their usefulness to calculus.
- 6.7: L'Hopital's Rule
- While this chapter is devoted to learning techniques of integration, this section is not about integration. Rather, it is concerned with a technique of evaluating certain limits that will be useful in the following section, where integration is once more discussed. This section introduces L'Hôpital's Rule, a method of resolving limits that produce the indeterminate forms 0/0 and ∞/∞.
- 6.8: Improper Integration
- When we defined the definite integral, we made two stipulations: The interval over which we integrated, [a,b], was a finite interval, and the function f(x) was continuous on [a,b](ensuring that the range of f was finite). In this section we consider integrals where one or both of the above conditions do not hold. Such integrals are called improper integrals.
Thumbnail: Graph showing a few iterations of Newton's method on the graph \(y=x^2\) with initial guess of \(x_0=4\). (Public Domain; Paul Breen).
Contributors and Attributions
Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/