ABSTRACT
The basic theory of integration presented in the last chapter provides a solid foundation for the analysis of integrals. Still, there are many problems, both practical and theoretical, whose resolution requires modifications or extensions of these ideas. This chapter addresses some of the more routine extensions: handling unbounded functions, unbounded intervals, and integrals which carry extra parameters. Riemann’s theory of integration works well for bounded functions
and bounded intervals, but many integrals arising in practice involve unbounded intervals or unbounded functions. Simple examples include∫ ∞
1 1 + x2
dx, (9.1)
1√ 1− x2 dx. (9.2)
Understanding when these integrals make sense involves the use of limits in a way that more or less parallels the study of infinite series. The second topic for this chapter is the study of functions defined
through integration. An example is the Laplace transform of a function f(x),
F (s) = ∫ ∞ 0
f(x)e−sx dx,
which converts certain problems of calculus or differential equations to problems of algebra. When functions are defined through integration, one would like to know when the function is differentiable, and how to calculate the derivatives.
