
14-6 Calculus – Differentiation and Integration
= + + +
a
c
k k
2
2
(sec tan log sec tan ) ,q q q q where is any constant
From
the figure give below, we have
sectan q q= =
+
=
cx
a
c x a
a
a
c
2 2 2
2
2
cc x a
a
cx
a
c x a
a
cx
a
k
x c x a
2 2 2 2 2 2
2 2 2
1
2
+
+
+
+
+
= + +
log
aa
c
cx c x a
a
c
a k
x c x a
a
c
cx c x a
2
2 2 2
2
2 2 2
2
2 2 2
2 2
1
2 2
log log
log
+ + − +
= + + + + ++C C, where is any constant.
cx
a
c x a
2 2 2
+
θ
Or
Put x
a
c
= sinh ,q
then
dx
a
c
d= cosh q q
c x a dx a a
a
c
d
a
c
d
a
2 2 2 2 2 2
2
2
2
+
∫
= +
∫
=
∫
=
sinh cosh
cosh
q q q
q q
22
2
2
1 2
2
1
2
2
2
2
2
c
d
a
c
d
a
c
a
cosh
( cosh )
sinh
q q
q q
q q
∫
= +
∫
= +
=
22
1
2
2 2c
cx
a
a
c
sinh sinh cosh
−
+ q q