Ex 4 p. 392 Integrating a Hyperbolic Function Let’s try some things for u u=1+sinh 2 x results in du = 2sinch(x)cosh(x)dx. No help. Identities? cosh 2 x-sinh 2 x = 1 Can you see power rule? How about now?
Unlike trig functions, hyperbolic functions are not periodic. Four of the six hyperbolic functions pass the horizontal line test, so their inverses will be functions. With two (hyperbolic cosine and secant) they are one-to-one if we restrict domains to positive real numbers. Since the hyperbolic functions can be written in terms of exponential functions, their inverses can be written in terms of natural log functions.
The hyperbolic secant can be used to define a curve called a tractrix or a pursuit curve Ex 5 p. 394 A tractrix A person is holding a rope that is tied to a boat. As the person walks, the boat travels along a curve called a tractrix, given by equation where a is the length of the boat. If a = 20 feet, find the distance the person must walk to bring the boat 5 feet from the dock. Solution: Notice that the length of y 1 is how far the person has walked. That is composed of Inserting y from our equation, When x = 5, this becomes Person
Ex 6 p. 395 More about a tractrix Show that the boat is always pointing toward the person in the setup from Ex. 5 For a point (x,y) on a tractrix, the slope of the graph gives the direction of the boat. (Refer to picture from Ex 5) The slope of line segment connect (0, y 1 ) with (x, y) is also this quantity (look at pink triangle) so it always points towards the person. It is because of this that it is also called the pursuit curve. Person
Ex 7 p. 395 Integration Using Inverse Hyperbolic Functions Let a = 3, u=2x, so du=2dx. This almost fits one of our new integration patterns.