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