Note that
(
)
3
4
d
x
x
dx
+
= 3x2 + 4x3 Therefore, an anti derivative of 3x2 + 4x3  is  x3 + x4 (iii)
We know that
1
1
1
(log )
0 and
[log (
)]
(
1)
0
d
d
x
,x
– x
–
,x
dx
x
dx
– x
x
=
>
=
=
<
Combining above, we get 
(
)
1
log
0
d
x
, x
dx
=x
≠
Therefore, 
1
log
dx
x
x
=
∫
 is one of the anti derivatives of 1
x Example 2 Find the following integrals:
(i)
3
2
1
x – dx
∫x
(ii)   
(32
1)
x
dx
+
∫
(iii)   ∫
23
1
(
2
–
)
+
∫
x
x
e
dx
x
Solution
(i)
We have
3
2
2
1
–
x –
dx
x dx –
x
dx
x
=
∫
∫
∫
(by Property V)
296
MATHEMATICS
= 
1
1
2
1
1
2
C
C
1 1
2
1
–
x
x
–
–
+
+




+
+




+
+




;  C1, C2 are constants of integration
= 
2
1
1
2
C
C
2
1
–
x
–x
–
–
+
 = 
2
1
2
1 + C
C
x2
–
x
+
= 
2
1 + C
2
x
x
+
, where C = C 1 – C2 is another constant of integration