ABSTRACT
Often, when attempting to solve a differential equation, one is lead to computing one or more integrals — after all, computationally, integration is the inverse of differentiation. Indeed, in the previous chapter was a second-order differential equation that was completely solved by repeated integration. The general case in which integration is immediately applicable is discussed in this chapter, as well as some practical aspects of using both the indefinite integral and the definite integral. In particular, guidelines for when a differential equation is “directly integrable” are laid out, along with conventions regarding the use of indefinite integrals that will simplify future computations. In addition, the possible advantages in using definite integrals are discussed and illustrated. These advantages are particularly evident when given an initial-value problem, when the functions being integrated are piecewise defined, and when there is no convenient formula for the integrated function. With regard to the last issue, there is further (brief) discussion in using numerical integration or “well-known” functions, such as the error function and the sine-integral function, which are defined by integrals that regularly arise in applications.
