The Chemistry Maths Book, Second Edition

6.6 Rational integrands. The method of partial fractions 179

Whenn 1 = 11 ,

Whenn 1 = 10 ,

(6.18)

This last integral is a standard integral that cannot be evaluated by the methods

described in this chapter (see Section 9.11).

The integralsI

n

are important in several branches of chemistry. For example,

in modern computational methods for the calculation of molecular wave functions,

the molecular orbitals are expressed in terms of ‘gaussian basis functions’. Such a

function is essentially the exponential multiplied by a polynomial, and the use

of these functions leads to integrals of the type discussed in this example.

0 Exercises 57–62

6.6 Rational integrands. The method of partial fractions

A rational algebraic function has the general formP(x) 2 Q(x)whereP(x)andQ(x)

are polynomials:

It was shown in Sections 2.6 and 2.7 that every such function can be expressed as the

sum of a polynomial and one or more partial fractions of types (if complex numbers

are excluded)

(6.19)

where nis a positive integer and the quadraticx

2

1 + 1 px 1 + 1 qhas no real roots; that is, its

discriminantp

2

1 − 14 qis negative.

Integrals of type (i)

These are the integrals that occur in the theory of elementary kinetic processes:

Z (6.20)

dx

xa

xa C n

nxa

C

n

n

()

()

()( )

• 
  

=

++ =

−

−+

• 
  

−

ln if

if

1

1

1

1

nn>















1

()

()

()

()

iii

1

2

xa

ax b

xpxq

nn

• 
  

,

• 
  

++

Px

Qx

aaxax ax

bbxbx b

n

n

m

()

()

=

++ ++

++ ++

01 2

2

01 2

2



 xx

m

e

−ar

2

Iedr

a

ar

0

0

2

1

2

==

−

Z

∞

π

Ierdr

a

e

ar ar

1

0

0

22

1

2

==−=

−−

































Z

∞ ∞

11

2 a

ZZ

00

2

22

1

2

1

∞∞

erdr

n

a

erdr I

n

ar n ar n

n

−−−

=

−

,=

() ()−

22

2

a

I

n−