Therefore,
2
2
dx
x
−x
∫
 =
2
1
dt
– t
∫
 = 
sin1
( )
C
–
t +
[by 7 4 (5)]
=
sin1
( –1)
C
–
x
+
Example 9 Find the following integrals :
(i)
2
6
13
dx
x
−x
+
∫
(ii)
32
13
10
dx
x
x
+
−
∫
(iii)
52
2
dx
x
x
−
∫
Solution
(i)
We have  x2 – 6x + 13 = x2 – 6x + 32 – 32 + 13 = (x – 3)2 + 4
So,
6
13
dx
x
2 −x
+
∫
 =
(
)
2
2
1
3
2
dx
x –
+
∫
Let
x – 3 = t Then dx = dt
Therefore,
6
13
dx
x
2 −x
+
∫
 =  
1
2
2
1 tan
C
2
2
2
–
dt
t
t
=
+
+
∫
[by 7 4 (3)]
=
1
1
3
tan
C
2
2
–
x –
+
312
MATHEMATICS
(ii)
The given integral is of the form 7