4  Integrals of Some Particular Functions
In this section, we mention below some important formulae of integrals and apply them
for integrating many other related standard integrals:
(1) ∫
2
2
1
–
=
log
+ C
2
+
dx–
x
a
a
x
a
x
a
308
MATHEMATICS
(2) ∫
2
2
1
+
=
log
+ C
2
–
dx–
a
x
a
a
x
a
x
(3) ∫
– 1
2
2
1 tan
C
dx
x
=
+
a
a
x + a
(4) ∫
2
2
2
2 = log
+
–
+C
–
dx
x
x
a
x
a
(5) ∫
– 1
2
2 = sin
+ C
–
dx
ax
a
x
(6) ∫
2
2
2
2 = log
+
+
+ C
+
dx
x
x
a
x
a
We now prove the above results:
(1)
We have  
2
12
1
(
) (
)
x – a
x
a
x – a
=
+
= 
1
(
) – (
)
1
1
1
2
(
) (
)
2
x
a
x – a
–
a
x – a
x
a
a
x – a
x
a


+


=




+
+




Therefore,  
2
2
1
2
dx
dx
dx
–
a
x – a
x
a
x – a


=


+


∫
∫
∫
= 
[
]
1
log (
)|
log (
)|
C
2
| x – a –
| x
a
a
+
+
= 1 log
C
2
x – a
a
x
a +
+
(2)
In view of (1) above, we have
2
2
1
1
(
)
(
)
2
(
) (
)
–
a
x
a
x
a
a
x
a
x
a
x


+
+
−
=


+
−

  = 1
1
1
2a
a
x
a
x


+


−
+


INTEGRALS         309
      Therefore,
2
–2
dx
a
x
∫
 = 
1
2
dx
dx
a
a
x
a
x


+


−
+


∫
∫
= 1 [ log |
|
log |
|]
C
2
a
x
a
x
a −
−
+
+
+
= 
1 log
C
2
a
x
a
a
+x
+
−
�Note  The technique used in (1) will be explained in Section 7 5 (3) Put x = a tan θ Then dx = a sec2 θ dθ