Then dx = a sec2θ dθ Therefore,
2
2
dx
x
+a
∫
 =
2
2
2
2
θ
θ
θ
sec
tan
a
d
a
+a
∫
  =
1
θ
θ
secθ θ = log (sec
tan )
C
d
+
+
∫
310
MATHEMATICS
=
2
1
2
log
1
C
x
x
a
a
+
+
+
=
2
1
log
log
C
x
x
a
|a|
2
+
+
−
+
=
2
log
C
x
x
a2
+
+
+
, where C = C1 – log |a|
Applying these standard formulae, we now obtain some more formulae which
are useful from applications point of view and can be applied directly to evaluate
other integrals (7)
To find the integral 
2
dx
ax
bx
c
+
+
∫
, we write
ax 2 + bx + c = 
2
2
2
2
2
4
b
c
b
c
b
a x
x
a
x
–
a
a
a
a
a








+
+
=
+
+


















Now, put 
b2
x
t
a
+
= so that dx = dt and writing 
2
2
42
c
–b
k
a
a
= ± We find the
integral reduced to the form 
2
2
1
dt
a
t
∫±k
 depending upon the sign of 
2
42
c
a–b
a






and hence can be evaluated