170 Chapter 6Methods of integration
EXAMPLES 6.7Integrals of type 3 :.
(i) , where ais an arbitrary number (buta 1 ≠ 1 − 1 ).
Letu 1 = 1 sin 1 x. Thendu 1 = 1 cos 1 x 1 dx, and
(ii).
Letu 1 = 1 sin 1 x. Thendu 1 = 1 cos 1 x 1 dx, and
0 Exercises 27, 28
Trigonometric and hyperbolic substitutions
The standard integrals listed in Table 6.3 can be evaluated by substituting appropriate
trigonometric and hyperbolic functions for the variable x.
IedueCe C
uu x==+=+Z
sinIe xdx
x=Z
sincos
Iudu
a
uC
a
xC
aa a==
+=
++Z
1
1
1
1
11sin
Ixxdx
a=Zsin cos
Zfx xdx()sin cos
Table 6.3
- a
2
x
2
- x
2
a
2
- a
2
x
2
Z
dx
ax
a
x
a
C
a
ax
ax
22
1
11
2
−=
+=
−
−
tanh ln
+C,
Z
dx
ax
a
x
a
C
22
1
1
=
+
−
tan
Z
dx
xa
x
a
Cxxa
22
122
=
+= + +
−
sinh ln ++C
Z
dx
xa
x
a
Cxxa
22
122
−
=
+= + −
−
cosh ln ++,C
Z
dx
ax
x
a
C
22
1
−
=
+,
−
sin