the Further Mathematics network
www.fmnetwork.org.uk
the Further Mathematics network
www.fmnetwork.org.uk
FP2 (MEI)
Inverse hyperbolic functions
Let Maths take you Further…
Inverse hyperbolic functions
Before you start:



You need to be confident in manipulating exponential and logarithmic functions.
You need to have covered the work on Maclaurin series from chapter 4.
You need to have covered Calculus from chapter 1 (integration using inverse trig
functions)
When you have finished…
You should:



Understand and be able to use the definitions of the inverse hyperbolic
functions.
Be able to use the logarithmic forms of the inverse hyperbolic functions.
Be able to integrate
1
and
x2  a2
1
x a
2
2
and related functions.
Notation
trig. functions
inverse trig.
functions
hyperbolic
trig. functions
inverse
hyperbolic
trig. functions
sin x
arcsin x
sinh x
arsinh x
cos x
arccos x
cosh x
arcosh x
tan x
arctan x
tanh x
artanh x
cosec x
arccosec x
cosech x
arcosech x
sec x
arcsec x
sech x
arsech x
cot x
arccot x
coth x
arcoth x
Latin for arc
Graphs
Use the graph of sinhx to sketch
the graph of arsinhx
Hint: use the
line y=x to help!
Remember for a function to have
an inverse it has to be a one-toone function
Sketch the graph of
arcoshx and state its
domain and range
The domain needs to be
refined to ensure the
function is one to one
Logarithmic form of the inverse
hyperbolic functions

y=arsinh x
so
x=sinh y
Summary
Differentiating inverse hyperbolic trig.
functions
Note: this can be done using the same technique that was used for
differentiating inverse trig. functions
y=arcosh x
x= cosh y
Results
We can now
integrate
expressions of
these forms!
We can also differentiate composite functions involving inverse hyperbolic
functions using the chain rule e.g. d
2
dx
ar sinh(2 x) 
( 2 x) 2  1
Using the previous results, together with the results we established by
considering inverse trig. Functions, we should now be able to integrate
functions of the form:
Inverse hyperbolic functions
When you have finished…
You should:



Understand and be able to use the definitions of the inverse hyperbolic
functions.
Be able to use the logarithmic forms of the inverse hyperbolic functions.
Be able to integrate
1
and
x2  a2
1
x a
2
and related functions.
2
Independent study:


Using the MEI online resources complete the
study plan for Hyperbolic functions 2
Do the online multiple choice test for this and
submit your answers online.