
12-26 Calculus – Differentiation and Integration
Example 12.37 Evaluate tan .
3
x dx
ò
Solution: The reduction formula for tan
n
x is
tan
tan
tan .
n
n
n
x dx
x
n
x dx
∫
=
−
−
∫
−
−
1
2
1
Thus,
tan
tan
tan
tan log sec .
3
2
2
2
1
2
x dx
x
x dx
x x c
∫
= −
∫
= − +
Example 12.38 Evaluate
sec
3
x dx
ò
and sec
4
x dx
ò
using reduction formula.
Solution: The reduction formula for sec
n
x is
sec
tan
sec .
n
n
n
x dx
x x
n
n
n
x dx
∫
=
−
+
−
−
∫
−
−
sec
2
2
1
2
1
Thus,
sec
sec
tan sec log sec tan
3
2
2
1
2
1
2
1
x dx
x dx
x x x x c
∫
= +
∫
= + + +
=
(tan sec log sec tan ).x x x x c+ + +
And
sec
tan sec
sec
tan sec
tan
tan (
4
2
2
2
3
2
3
3
2
3
1
3
x dx
x x
x dx
x x
x c
x
∫
= +
∫
= + +
= ssec ) .
2
2x c+ +
12.5.2 Integrals of the Form
dx
a b nx
dx
a b nx
dx