In the main text we found that the inverse hyperbolic functions, hyperbolic arcsine
and hyperbolic arccosine, can appear as part
of the integrals of certain quadratic-under-the-radical type of integrands. Here
you have an opportunity to get more familiar with these functions. We also found
a useful way of expressing the related function, hyperbolic arctangent,
in terms of logs.
Hyperbolic Arctangent
We have already seen that with a little algebra we can derive
arctanh(x) =
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1
ln
2
|
|
1 + x
1 - x
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and we observed that this has as its domain only the interval,
-1 < x < 1. At
x = 1
the above formula gives you a division by zero. At
x = -1
it gives you the log of zero. And everywhere else outside the domain it gives you
the log of a negative number.
To illuminate what hyperbolic arctangent is all about, we take its
derivative. This is easier if we apply the log-of-a-quotient identity
first.
1 1
arctanh(x) = ln(1 + x) - ln(1 - x) eq. 11.6b-1
2 2
When we take the derivative of that we get
darctanh(x)
=
dx
|
1 1
+
2(1 + x) 2(1 - x)
|
which we can put over a common denominator to
get
darctanh(x)
=
dx
|
1 - x + 1 + x
=
2(1 + x)(1 - x)
|
1
eq. 11.6b-2
1 - x2
|
Clearly the domain of a function's derivative cannot extend beyond the
domain of the original function. But in this case, the expression we have
arrived at for the derivative of
arctanh(x) does appear
to be extensible to a larger domain that includes all real numbers except
x = 1 and
x = -1.
The question is, can we extend the definition of the
hyperbolic arctangent
itself to include
x greater than
1 and
x less than
-1?
Recall that when we looked at finding an antiderivative of 1/x,
we discovered that ln(x) worked just fine for positive x,
but for negative x, we had to take the absolute value before applying
the log. The same thing works for finding an antiderivative of
1/(1 - x2). So for an extended definition
of hyperbolic arctangent, let
arctanh(x) =
|
1
ln
2
|
|
1 + x
1 - x
|
|
1 1
= ln|1 + x| - ln|1 - x| eq. 11.6b-3
2 2
|
This too has a domain of all real numbers except
x = ±1.
At all other points, its derivative is
exactly
darctanh(x)
=
dx
|
1
1 - x2
|
The extended domain makes our new definition of
hyperbolic arctangent
much more satisfying and useful than just the inverse function of
hyperbolic tangent.
Plot of Extended arctanh(x)
|
Here is a plot of the extended definition of arctanh(x) that we
developed above. That plot is shown in blue. The green function is its
derivative. Since arctanh(x) is an odd function,
its derivative is an even function.
Observe the behavior of each of these functions near the
points x = ±1.
Look also at the blue plot between plus and minus 1. Observe
that if you turned that on its side and took its mirror image, you would
have the exact plot of hyperbolic tangent that you saw in the
main text, indicating that this branch is indeed the inverse of
hyperbolic tangent.
This extended version of hyperbolic arctangent can be used for
integrating functions that have a quadratic with real roots in the
denominator, but no radical. There is, however, another way of doing
these, which you will see in the section on partial fractions. And
although using hyperbolic arctangent is just as valid a method
for integrating such functions,
most instructors will prefer you use the partial fraction method.
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Hyperbolic Arccosine
As we observed from the main text, using the formula
arccosh(x) = arctanh
|
|
______
√x2 - 1
x
|
|
already gives
hyperbolic arccosine a wider domain than it would
have if you defined it strictly as the inverse of the
hyperbolic cosine
function. If you look at the
graph, you can
see that even using the inverse function definition, you would still have
to restrict that to only the first quadrant branch of
hyperbolic cosine.
You can simplify the log-expression for hyperbolic arccosine simply by
using a little algebra. We start with expanding it in terms of the hyperbolic arctangent.
1
arccosh(x) = ln
2
|
|
______
√x2 - 1
1 +
x
|
|
1
- ln
2
|
|
______
√x2 - 1
1 −
x
|
|
eq. 11.6b-4a
|
From there, using log identities and algebra, we get
1
ln
2
|
|
______
√x2 - 1
1 +
x
|
|
1
- ln
2
|
|
______
√x2 - 1
1 −
x
|
|
1
= ln
2
|
|
______
x + √x2 - 1
______
x - √x2 - 1
|
|
=
|
1
ln
2
|
|
x2 - (x2 - 1)
______
(x - √x2 - 1)2
|
|
= ln
|
|
1
______
x - √x2 - 1
|
|
= ln
|
|
______
x + √x2 - 1
x2 - (x2 - 1)
|
|
=
|
______
ln|x + √x2 - 1| eq. 11.6b-4b
|
To take the derivative of the simplified expression, use
the chain rule.
darccosh(x)
=
dx
|
|
1
______
x + √x2 - 1
|
|
|
x
1 + ______
√x2 - 1
|
|
1
= ______ eq. 11.6b-5
√x2 - 1
|
In the right-hand pair of parentheses, put the 1
over a common denominator with the fraction and then add. You will see
then that the numerator in the right-hand pair of parentheses cancels
with the denominator in the left-hand pair of parentheses.
Even though the inverse of the hyperbolic cosine does not include
in its domain any x less than 1, this derivative is
easily extensible to include values of x that are less than
or equal to -1. Our formula for using the hyperbolic arctangent
to generate the hyperbolic arccosine also extends to this domain.
Observe that the derivative, as we have extended it, is an even function.
Hence the extended definition of hyperbolic arccosine must be an odd
function.
Plot of Extended arccosh(x)
|
Here is a plot of the extended definition of arccosh(x) that we
developed above. That plot is shown in blue. The green function is its
derivative.
Observe the behavior of each of these functions near the
points x = ±1. Notice also that
neither the function nor its derivative are defined in the interval
-1 < x < 1.
It is a bit startling that with hyperbolic cosine
being an even function, this extended definition of its
inverse should be an odd function. But that is the way
it works out.
The extended version of hyperbolic arccosine often appears in the integrals of
functions that have a quadratic with real roots
under the radical, and where the x2 term is positive.
Such functions will always have two branches. With the definition
of hyperbolic arccosine extended as shown here, integral solutions
that contain this function will always apply to both branches of
the function.
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|
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Hyperbolic Arcsine
This one is well-behaved compared with the other two. That is, the inverse
of the hyperbolic sine naturally has as its domain all real numbers.
It is also continuous and differentiable everywhere. Hence there is no need
to extend it. Based upon what we've
already developed on this page, it should be clear
that
1
arcsinh(x) = ln
2
|
|
x
1 + ______
√x2 + 1
|
|
1
- ln
2
|
|
x
1 - ______
√x2 + 1
|
|
eq. 11.6b-6a
|
Observe that you don't even need the absolute value signs here because
the expressions inside the logs are always positive.
This one also simplifies with the help of log identities and some algebra
1
ln
2
|
|
x
1 + ______
√x2 + 1
|
|
1
- ln
2
|
|
x
1 - ______
√x2 + 1
|
|
1
= ln
2
|
|
______
√x2 + 1 + x
______
√x2 + 1 - x
|
|
=
|
1
ln
2
|
|
(x2 + 1) - x2
______
(√x2 + 1 - x)2
|
|
= ln
|
|
1
______
√x2 + 1 - x
|
|
= ln
|
|
______
√x2 + 1 + x
(x2 + 1) - x2
|
|
=
|
______
ln|√x2 + 1 + x| eq. 11.6b-6b
|
I am going to let you take the derivative of this one yourself for practice.
Look at how I did the derivative of the
hyperbolic arccosine and do
this one analogously. In the end you should
get
darcsinh(x) 1
= ______ eq. 11.6b-7
dx √x2 + 1
|
Plot of arcsinh(x)
|
Here is a plot of arcsinh(x).
There's not a whole lot to say about it because it is so well-behaved.
Hyperbolic arcsine often appears in the integrals of functions that
have a quadratic under the radical that has no real roots.
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