6 2 Integrals of some more types
Here, we discuss some special types of standard integrals based on the technique of
integration  by parts :
(i)
2
2
x
a
dx
−
∫
(ii)
2
2
x
+a dx
∫
(iii)
2
2
a
−x dx
∫
(i)
 Let 
2
2
I
x
a dx
=
−
∫
Taking constant function 1 as the second function and integrating by parts, we
have
I =
2
2
2
2
1
2
2
x
x
x
a
x dx
x
a
−
−
−
∫
=
2
2
2
2
2
x
x
x
a
dx
x
a
−
−
−
∫
 = 
2
2
2
2
2
2
2
x
a
a
x
x
a
dx
x
a
−
+
−
−
−
∫
INTEGRALS         329
=
2
2
2
2
2
2
2
dx
x
x
a
x
a dx
a
x
a
−
−
−
−
−
∫
∫
=
2
2
2
2
2
I
dx
x
x
a
a
x
a
−
− −
−
∫
or
2I =
2
2
2
2
2
dx
x
x
a
a
x
a
−
−
−
∫
or
I = ∫
2
2
x – a dx = 
2
2
2
2
2
–
–
log
+
–
+ C
2
2
x
a
x
a
x
x
a
Similarly, integrating other two integrals by parts, taking constant function 1 as the
second function, we get
(ii) ∫
2
2
2
2
2
2
2
1
+
=
+
+
log
+
+
+ C
2
a2
x
a dx
x
x
a
x
x
a
(iii) ∫
2
2
2
2
2
–1
1
–
=
–
+
sin
+ C
2
a2
x
a
x dx
x a
x
a
Alternatively, integrals (i), (ii) and (iii) can also be found by making trigonometric
substitution x = a secθ in (i), x = a tanθ in (ii) and x = a sinθ in (iii) respectively Example 23 Find 
2
2
5
x
x
dx
+
+
∫
Solution Note that
2
2
5
x
x
dx
+
+
∫
 =
2
(
1)
4
x
dx
+
+
∫
Put  x + 1 = y, so that dx = dy Then
2
2
5
x
x
dx
+
+
∫
 =
2
22
y
dy
+
∫
=
2
2
1
4
4
log
4
C
2
2
y
y
y
y
+
+
+
+
+
        [using 7