Hyperbolic Functions - LTCC Online

Hyperbolic Functions

Definition of the Hyperbolic Functions

We define the hyperbolic functions as follows:

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

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

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