The Chemistry Maths Book, Second Edition

6.6 Rational integrands. The method of partial fractions 183

The general form:

The numerator can be written as

(6.28)

so that the integral is expressed in terms of the special cases discussed above:

(6.29)

0 Exercises 70, 71

Rational trigonometric integrands

By trigonometry,

(6.30)

Then, dividing the numerators and denominators bycos

2

1 θ 22 and puttingt 1 = 1 tan 1 θ 22 ,

we obtain

(6.31)

The trigonometric functionssin 1 θandcos 1 θare rational functions of t. A trigonometric

function of θthat becomes a rational (algebraic) function of twhen we make the

substitutiont 1 = 1 tan 1 θ 22 is called a rational trigonometric functionof θ. Every such

function can be integrated by the methods described earlier in this section. Thus, if

the integrand in the integral is a rational trigonometric function of θ, the

substitution

td (6.32)

t

=, =dt

• 
  

tan

θ

θ

2

2

1

2

Zfd()θ θ

sinθθcos tan

θ

=

• 
  

,=

−

• 
  

,=

2

1

1

1

2

2

2

2

t

t

t

t

t

cos cos sin

cos sin

cos sin

θ

θθ

θθ

θ

=−=

−

• 
  

22

2

2

2

2

2

2

2

22

θθ

2

sin sin cos

sin cos

cos sin

θ

θθ

θθ

θθ

==

• 
  

2

22

2

22

2

2

2

2

+−













++

b

ap dx

xpxq

n

2

2

Z

()

ZZ

ax b

xpxq

dx

axp

xpxq

dx

nn

• 
  

++

=

• 
  

()()++

22

2

2

ax b

a

xp b

ap

+= +

()

+−













2

2

2

Z

ax b

xpxq

dx

n

• 
  

()++

2