Inverse Hyperbolic Functions and Integrals Leading to Them

Abstract

Common integration is only the memory of differentiation…the different artifices by which integration is effected, are changes, not from the known to the unknown, but from forms in which memory will not serve us to those in which it will. — Augustus de Morgan In this chapter the inverse hyperbolic functions are briefly reviewed before various standard types of integrals that lead to inverse hyperbolic functions are considered. § The Inverse Hyperbolic Functions From Chapter 9 you may recall that since the functions sinh and tanh are both increasing functions on their domain, both are one-to-one functions and accordingly will have well-defined inverses. Those inverses are denoted by sinh -1 x and tanh -1 x, respectively. The function cosh, however, is not one-to-one on its domain. To obtain an inverse its domain must be restricted. By convention the domain of the hyperbolic cosine function is restricted to the interval. Since the hyperbolic functions are defined in terms of the exponential function, one would expect it would be possible to write each of the inverse hyperbolic functions in terms of the logarithmic function, the logarithmic function being the inverse of the exponential function. This is indeed possible. For example, to find an expression for the inverse hyperbolic sine function in terms of the logarithmic function, set y = sinh -1 x, then sinh y = x, or After multiplying throughout by 2e y and rearranging, the following equation results: which is quadratic in e y . On applying the quadratic formula, solving for e y gives Here, since e y 0, the positive square root is taken. Taking the natural logarithm of both sides of the equation gives Inverses for the other five hyperbolic functions can be found in a similar manner. These, together with each function's associated domain and range, are summarised in Table 13.1. In the table, where a plus and minus sign appear, the positive sign is used for x > 0, while the negative sign is used for x Graphs for all six inverse hyperbolic functions are shown in Figure 13.1. Since the inverse hyperbolic functions can be expressed in terms of logarithms, their derivatives can be readily found.