
14-26 Calculus – Differentiation and Integration
Let x a t x
2 2 2 2
− = .
Then x
a
t
x dx
a
t
t dt
2
2
2
2
2 2
1 1
=
−
⇒ =
−( )
.
Thus,
x
x a
dx
x
t x
a
t
t dt
a
x t t
2
2 2 3 2 2 2 3 2
2
2 2
2
2 2 2
1
1
1
( ) ( ) ( )
( )
/ /
−
∫
=
−
∫
=
−
22
2
2
2 2 2 2
2 2
2 2
1 1
1
1
1
1 1
1
dt
a
t
a t t
dt
t t
dt
t t
∫
=
−
−
∫
=
−
∫
= +
−
( )
( )
∫
= +
−
∫∫
= − −
−
+
+
= −
−
+
−
dt
t
dt
t
dt
t
t
t
c
x
x a
x x
1 1
1
1 1
2
1
1
1
2
2 2
2 2
2
log
log
−−
+ −
+
a
x x a
c
2
2 2
.
S u m mARy
• There are various techniques to integrate the irrational function.
• Irrational function can be integrated by the techniques of trigonometric transformations.
• There is a method to deal with the integrals of the form ax bx c
2
+ + ,
1
2
ax bx c+ +
and
Ax B
ax bx c
+
+ +
2