Hyperbolic Functions

Definition of the Hyperbolic Functions

We define the hyperbolic functions as follows:

 ex - e-x      sinh x =                                             2                       ex + e-x      cosh x =                                             2                         sinh x      tanh x =                                      cosh x

Properties

1. (cosh x)2 - (sinh x)2 = 1

2. d/dx(sinh x) = cosh x

3. d/dx(cosh x) = sinh x

Proof of A

We find

The Derivative of the Inverse Hyperbolic Trig Functions

Proof of the third identity

We have

tanh(arctanh x)  =  x

Taking derivatives implicitly, we have

d
sech2(arctanh x)          arctanh x  =  1
dx

Dividing gives

d                                             1
arctanh x       =
dx                                 sech2(arctanh x)

Since

cosh2(x) - sinh2(x) = 1

dividing by cosh2(x), we get

1 - tanh2(x) = sech2(x)

so that

d                                       1                           1
arctanh x  =                                   =
dx                         1 - tanh2(arctanh x)          1 - x2

Integration and Hyperbolic Functions

Now we are ready to use the arc hyperbolic functions for integration

Example:

Example

Evaluate

Solution

Although this is not directly a derivative of a hyperbolic trig function, we can use the substitution

u = x2 ,    du = 2x dx

To change the integral to