The Chemistry Maths Book, Second Edition

176 Chapter 6Methods of integration

Then, solving for I,

and

0 Exercises 49–51

6.5 Reduction formulas

The method of integration by parts can be used to derive formulas for families of

related integrals. Consider the integral

where nis a positive integer. Choosingu 1 = 1 x

n

and in the formula (6.14),

or

(6.15)

This result is called a reduction formulaforI

n

, or a recurrence relationbetweenI

n

andI

n− 1

. For example

where

Then

I

a

ex

a

x

a

x

a

C

ax

3

32

23

1366

= −+−

















• 
  

Iedx

a

eC

ax ax

0

1

==+Z

I

a

xe

a

II

a

xe

a

II

a

xe

ax ax ax

3

3

22

2

11

13 12 11

=−,=−,=−

aa

I

0

I

a

xe

n

a

I

n

nax

n

=−

−

1

1

I

a

xe

n

a

xedx

n

nax n ax

=−

−

1

1

Z

d

dx

e

ax

v

=

Ixedx

n

nax

=Z

Z

0

2

1

∞

exdx

a

a

−ax

=

• 
  

cos

Ie xdx

a

exaxC

ax ax

==

• 
  

−+

−−

Z cos (sin cos )

1

1

2