184 Chapter 6Methods of integration
gives
(6.33)
wheref(θ)in terms of tis a rational function of t.
0 Exercise 72
EXAMPLE 6.21Examples of the substitutiont 1 = 1 tan 1 θ 22.
(i)
(ii)
making use of standard integral 5 in Table 6.3.
0 Exercises 73–75
This method can be applied in all cases but is not always the simplest in practice.
For example, the application of the method to the integration of the elementary
trigonometric functions sin 1 θand cos 1 θis considerable more complicated than the
use of the standard integrals.
6.7 Parametric differentiation of integrals
Consider the indefinite integral
The integral can be treated as a function of the parameter α; differentiation then
gives
d
d
edx
d
d
eC
x
e
xxααα
αα
ααZ
−−=− +
=+
11
2−−αxZedx e C
−−xx=− + , ≠
ααα
α
1
() 0
=
−
1
4
22
22
ln
tan
tan
θ
θ
C
=
−
=
−
Z +
dt
t
t
t
C
4
1
4
2
2
2ln
ZZ
d
t
t
t
θ
35 θ
2
1
35
1
1
222=
−
cos
=
−
dt
t
Z dt
2
82
2ZZ Z
d
t
t
t
dt
t
d
θ
sinθ
=
=
2
1
2
1
1
22tttC=+=ln ln tanθ 2 +C
ZZfd
f
t
() dt
()
θθ
θ
=
2
1
2