182 Chapter 6Methods of integration
EXAMPLE 6.19Integrate.
The quadratic functionx
21 + 12 x 1 + 15 can be written as(x 1 + 1 1)
21 + 12
2. Then, by equation
(6.27), witha 1 = 12 andu 1 = 1 x 1 + 11 ,
0 Exercise 68
EXAMPLE 6.20Integrate.
By equation (6.26), withn 1 = 13 ,a 1 = 12 ,andu 1 = 1 x 1 + 11 ,
wheretan 1 θ 1 = 1 u 2 a 1 = 1 (x 1 + 1 1) 22. Then, by reduction as in Example 6.14,
We need to change variable from θback to u(and then to x). Iftan 1 θ 1 = 1 u 2 athen
θ 1 = 1 tan
− 11 (u 2 a)and it is readily verified that
Then
and
0 Exercise 69
Z
dx
xx
x
xx
x
() x
()
()
()
(
232225
1
32
21
25
31
++ 4
=
++
22125
3
8
1
2
++
−x
x
C
)
tan
Zcos
()
tan
43222 2211
4
3
8
3
8
θθd
ua
ua
ua
ua
u
a
=
−
+C
sinθθ= cos
,=
u
ua
a
ua
22 22=+++
1
4
3
8
3
8
3sin cosθθ θθθsin cos C
ZZcos sin cos cos
4321
4
3
4
θθdd=+θ θ θθ
ZZ
dx
xx
d
()
cos
23425
1
32
++
= θθ
Z
dx
()xx
23++ 25
ZZ
dx
xx
dx
x
x
C
222125 1 2
1
2
1
2
++
=
++
=
−()
tan
Z
dx
xx
2++ 25