
Methods of Integration 12-11
Solution:
a. Let x = at, then dx = a dt
dx
x a
a dt
a t
a
dt
t
a
t
a
x
a
2 2 2 2
2
1
1
1
1
1
1
1
+
=
+
∫∫
=
+
∫
=
=
−
−
( )
tan ( )
tan
b.
x a
x a x a
a
x a x a
x a x a
dx
a x a
2 2
1
2
1
2
1
−
=
− +
∫∫
=
+ − −
− +
∫
=
−
( )( )
( ) ( )
( )( )
−−
+
∫
= − − +
=
−
+
1
1
2
1
2
x a
dx
a
x a x a
a
x a
x a
(log( ) log( ))
log
Similarly, we obtain
dx
a x a
dx
x
a
a
x
a
d
x
a
2 2 2 2
2
1
1
1 1
1
−
=
−
∫∫
=
−
∫
−
∫
=
=
−
,
tanh
which is in the form
1
1
1
2
1
x
dx
a
x
a
11
1
a
x
a
coth .
−