Calculus: Analytic Geometry and Calculus, with Vectors

508 Exponential and logarithmic functions

(9.36) sinh-1 x = log (x + x'- T-1)' d sinh-1 x =^1

dx 1 +x2

1

(9.361) cosh-1 x = log (x + dz cosh-1 x = 7X:2=-7

(x > 1)

(9.362) tanh-1 x =. log1 + x,

d

tank-1 x = 1

1 -x dx 1 -x2

OxI < 1)

(9.363) coth-1 x = i log

x + d coth-1x = -1

x-1 dx x2-1

(9.364)

(9.365)

(lxI > 1)

1 + 1 - xz d

sech-1 x = log , sech-1 x = -1

x dx x 1 -x2

(0<x<1)

csch-1 x = log 1 + 1

-+X 2

, d csch-1x = -1

x dx x 1 +x2

(x > 0)

Because of the growing tendency to use exponentials to eliminate

calculations involving hyperbolic functions and even trigonometric func-

tions, it seems unwise to devote more time to the formal aspects of the

subject. It is, however, of interest to know how hyperbolic functions

are related to hyperbolas. Setting

t -t t - t

(9.37) x=cosh t=e

e -t

y=sinht=e

2 2e

we see that x > 0 and x2 - y2 = 1, so P(x,y) lies on the branch of the

rectangular hyperbola shown in Figure 9.371. A reasonable way to try

Figure 9.371 Figure 9.372