Then dx = a cosθ dθ Therefore,
 
2
2
dx
a
−x
∫
 =
2
2
2
θ θ
θ
cos
sin
a
d
a – a
∫
=
θ = θ + C = sin1
C
–
x
d
a
+
∫
(6) Let x = a tanθ 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