Chapter 11 Computing integrals
All continuous functions have integrals, but not all of the integrals have clean familiar formulas. For example, the definite integral \(\displaystyle\int_3^8 \frac{1}{\ln x} \, dx\) is a well defined quantity; indeed \(\displaystyle\int_a^b \frac{1}{\ln x} \, dx\) is well defined for any \(b > a > 1\text{,}\) but the function \(b \mapsto \displaystyle\int_a^b (1 / \ln x) \, dx\) is not equal to any combination of named functions such as powers, logs, exponentials and trig functions. The same is true of the normal (bell curve) density function \(e^{-x^2}\text{,}\) or \(\sqrt{\sin x}\) or \(\sqrt{1-4x^2} / \sqrt{1-x^2}\text{.}\) The prevalence of functions like this is the reason we need good numeric approximations to integrals (as discussed in Unit 10.2 and Unit 10.3). In the remainder of this section we concentrate on anti-derivatives for which reasonably nice exact expressions exist.
