Integration of hyperbolic and inverse hyperbolic

functions

Submitted By

Vikram Kumar (maths)

P.G.G.C for Girls

Sec – 11, Chandigarh.

• Integration of hyperbolic

• Inverse hyperbolic functions

• Reduction formulae

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

1.

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

Learning outcomes

Integrate by parts and generate a reduction formula

Integrate by parts using a reduction formula

Evaluate integrals with integrands of the form sinnx and cosnx using reduction

formulas