calculus of hyperbolic functions - pbworks
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Osborn’s rule
From previous examples we can see that the close comparison
between identities in trigonometric functions and hyperbolic
functions can be converted into a formulae known as Osborn’s rule,
which states that the cos should be converted to cosh and sin
converted to sinh, except when there is a product of two sines, we
must change the sign.
1sincos 22 xx 1sinhcosh 22 xx
However, whenever using Osborn’s rule care must be taken as the
product of two sines is sometimes disguised egx
xx
2
22
cos
sintan
Calculus of Hyperbolic Functions
If xx eexxf 2
1cosh
then xeexf xx sinh2
1'
xxdx
dsinhcosh
If xx eexxf 2
1sinh
then xeexf xx cosh2
1'
xxdx
dcoshsinh
Similarly
From this it follows that
and
cxxdx coshsinh
cxxdx sinhcosh
Example
Differentiate
(a)
(b)
xtanh
xcosech
Example
Find the derivative of cosh3x and evaluate dxx5.0
0
3sinh
Example
Find (a)
(b)
dxx tanh
dxx3sinh
Example
Integrate with respect to x xex cosh
If then
Inverse Hyperbolic functions
The inverse hyperbolic functions are defined in a similar manner to
the inverse of trigonometric function.
xy sinhIf then yx 1sinh
xy cosh yx 1cosh
The graphs of inverse hyperbolic functions are obtained from those of
the hyperbolic functions by interchanging the x and y axes.
0 x
y
0 x
yxy 1sinh xy 1cosh
This function is a one to one function This function is a one to many function
Example
Let so
a) Express in the terms of and hence show that
b) Deduce that
xy 1tanh yx tanh
ytanh ye
x
xe y
1
12
x
xx
1
1ln
2
1tanh 1
Derivatives of inverse Hyperbolic functions
Example
Find the derivative of with respect of x.
a
x1sinh
Example
Find the derivative of with respect of x.
a
x1cosh
1
1sinh
2
1
xx
dx
d
1
1cosh
2
1
xx
dx
d
Example
Differentiate
(i)
(ii)
12cosh 1 x
x
1sinh 1
1
1cosh
1
1sinh
sechtanh
sinhcosh
coshsinh
2
1
2
1
2
xx
xx
xx
xx
xxdx
dyy
Use of Hyperbolic functions in integration
Example
Using the previous results write down the values of
(i)
(ii)
dxx 1
1
2
dxx 1
1
2
Example
a) Differentiate with respect to x
b) Hence find dxx 9
1
2
3sinh 1 x
Example
Use the substitution of to show thatux cosh2
cx
dxx
2cosh
4
1 1
2
In general
ca
xdx
ax
ca
xdx
ax
1
22
1
22
cosh1
sinh1
Example Evaluate 13 2x
dx
Example
a) Express in the form where A, B
and C are constants.
b) Evaluate in terms of natural logarithms
584 2 xx CBxA 2
dxxx
7
4 2 584
1
Example
Evaluate
Leaving your answer in terms of natural logarithms.
1
3
2 136 dxxx
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