
Methods of Integration 12-21
Example 12.30 Evaluate sech x dx
0
∞
∫
.
Solution: We know that
sech x
e e
e
e
x x
x
x
=
+
=
+
−
2 2
1
2
∴
∫
=
+
∫
∞ ∞
sech x dx
0
2
0
2
1
e
e
dx
x
x
Let u e du e dx
x x
= ⇒ =
∴
∫
=
+
∫
=
=
=
∞ ∞
−
∞
−
∞
sech x dx
0
2
0
1
0
1
0
2
1
2
2
2
u
du u
e
x
tan
tan
[
ttan tan ]
[tan tan ]
− ∞ −
− −
−
= ∞−
= −
= =
1 1 0
1 1
2 1
2
2 4
2
4 2
e e
p p p p
..
12.5 INTEGRATION OF TRIGONOMETRIC FUNCTIONS AND THEIR
REDUCTION FORMULAE
In this section, we shall discuss the integration of various trigonometric functions using their reduction
formulae.
Integration of sin
n
x where n is a Positive Integer
There are two cases accordingly whether n is odd or even:
• If n is an odd positive integer, the ...