integration of hyperbolic and inverse hyperbolic...
TRANSCRIPT
Integration of hyperbolic and inverse hyperbolic
functions
Submitted By
Vikram Kumar (maths)
P.G.G.C for Girls
Sec – 11, Chandigarh.
Definitions of Hyperbolic functions
sinh2
x xe ex
cosh
2
x xe ex
2 2
2 2cosh sinh 12 2
x x x xe e e ex x
sinhtanh
cosh
x x
x x
x e ex
x e e
coshth
sinh
x x
x x
x e eco x
x e e
1 2sech
cosh x xx
x e e
1 2csch
sinh x xx
x e e
Generating a reduction formula
A reduction formula is a formula
which connects a given integral with
another integral in which the integrand
is of same type but of lower degree or
order.
Generating a reduction formula
Using the integration by parts formula:
it is easily shown that:
udv uv vdu
1n x n x n xx e dx x e n x e dx
Generating a reduction formula
Writing:
then
can be written as:
This is an example of a reduction formula.
n x
nI x e dx
1n x n x n xx e dx x e n x e dx
1
n x
n nI x e nI
Programme 17: Reduction formulas
Generating a reduction formula
Sometimes integration by parts has to be repeated to obtain the
reduction formula. For example:
1
1 2
1
2
cos
sin sin
sin cos ( 1) cos
sin cos ( 1)
n
n
n n
n n n
n n
n
I x xdx
x x n x xdx
x x nx x n n x xdx
x x nx x n n I
Definite integrals
When the integral has limits the reduction formula may be simpler.
For example:
0
1
20
1
2
cos
sin cos ( 1)
( 1)
n
n
x
n n
nx
n
n
I x xdx
x x nx x n n I
n n n I
Integrands of the form and sinn x cosn x
The reduction formula for is
and . . .
sinn
nI xdx
1
2
1 1sin .cosn
n n
nI x x I
n n
Integrands of the form and sinn x cosn x
the reduction formula for is:
These take interesting forms when evaluated as definite integrals
between 0 and /2
cosn
nI xdx
1
2
1 1cos .sinn
n n
nI x x I
n n
Integrands of the form and sinn x cosn x
The reduction formulas for are both:
where
(a) If n is even, the formula eventually reduces to I0 = /2
(b) If n is odd the formula eventually reduces to I1 = 1
/ 2
0
sinn
nI xdx
2
1n n
nI I
n