Identities
Derivatives and Integrals
Useful Inverse Identities
Inverse Derivatives and Integrals
Inverse Hyperbolic Functions as Natural Logarithms
Hyperbolic Functions
There are six hyperbolic functions:
These functions can be used to describe the motion of waves in elastic solids,
the shape of hanging cables, and the temperature distributions in metal cooling fins.
Note that the definitions of tanh, coth, sech, and csch, match the definitions of the
corresponding trigonometric functions, and identities are similar.
Identities
Here are other examples:
.The proofs consist of straightforward algebra.
Derivatives and Integrals
Notice the similarities to the trigonometric functions.
See Example 1, page 522.
Inverse Hyperbolic Functions
Useful Identities
These identities can be used to calculate the values of the inverse functions
on the left side of the equation on calculators that give only the values of the
inverse functions on the right side of the equation.
Derivatives and Integrals.
See Example 2, page 525.
See Example 3, page 526.
The Inverse Hyperbolic Functions as Natural Logarithms
Given that
,we want to solve the equation for
.Symbolically, the solution would be the
inverse hyperbolic sinh function
.But we want an explicit solution, so proceed to clear the equation of fractions:
This is a quadratic equation in
.
Put it in standard form:
.Using the quadratic formula, solve the equation for
:
.But
and
is
always positive, so we must throw away the negative sign - otherwise the difference will be negative - and retain only the positive sign:
.Take the natural logarithm of both sides of the equation:
.So
or
.This already is the inverse function we want, but we need to write it
in the conventional way. That is, we want to write the inverse function
in terms of
.
In the above formula, just replace 
with
:
.Formulas for the other inverse hyperbolic functions can be derived in the same way.
next Basic Integration Formulas Module 4
Top Integration Techniques
L'Hopital's Rule
Improper Integrals