
Integration of Irrational Functions 14-11
=
∫
=
∫
= +
∫
= −
1
81
1
81
1
81
1
1
81
4 2 2
2 2
csc csc csc
(cot ) csc
(co
q q q q q
q q q
d d
d
tt )( csc )
(cot ) (cot ) ,
2 2
2 2
1
1
81
1 1
q q q
q q q
+ −
∫
= − + +
∫
d
d d
which is in
tthe form
From figure
( )
cot
cot
x dx
c
2
3
1
1
81 3
+
∫
= − +
+
q
q
we havecot q =
−
= −
−
+
−
9
1
81
1
3
9 9
2
2
3
2
x
x
x
x
x
x
+c.
14.3 INTEGRATION OF IRRATIONAL FUNCTIONS OF THE FORM
ax bx c
ax bx c
2
2
AND
1
+ +
+ +
In this section, we describe a way of dealing with the integrals of the form ax bx c
2
+ + and
1
2
ax bx c+ +
.
In order to handle the general quadratics like ax
2
+ bx + c, we use the process of “completing the
square” to ...