Hyperbolic Function

Find the area under the curve y = -2x^-1 from x = -5 to x = -1. (See Figure 4.3.5.)

Figure 4.3.5

The area is given by the definite integral

First compute the indefinite integral

Now compute the definite integral.

This example illustrates the need for the absolute value in the integration rule

x^-1 dx = ln|x| + C.

The natural logarithm ln x is undefined at x = - 5 and x = - 1, but ln |x| is defined for all x 0. The absolute value sign is put in when integrating x^-1 and removed when differentiating ln |x|.