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

 

 
               Theorem

    

 

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

 

For the derivative of the inverse sech(x) click here

 



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

       

 



Back to Math 105 Home Page

Back to the Math Department Home

e-mail Questions and Suggestions