Chapter 40
Integration using
trigonometric and
hyperbolic substitutions
40.1 Introduction
Table 40.1 gives a summary of the integrals that require
theuse oftrigonometric and hyperbolic substitutions
and their application is demonstrated in Problems 1
to 27.
40.2 Worked problems on integration
ofsin^2 x,cos^2 x,tan^2 xandcot^2 x
Problem 1. Evaluate
∫ π
4
0
2cos^24 tdt.
Since cos2t=2cos^2 t−1 (from Chapter 17),
then cos^2 t=
1
2
( 1 +cos 2t)and
cos^24 t =
1
2
( 1 +cos 8t)
Hence
∫ π
4
0
2cos^24 tdt
= 2
∫ π
4
0
1
2
( 1 +cos 8t)dt
=
[
t+
sin8t
8
]π
4
0
=
⎡
⎢
⎣
π
4
+
sin8
(π
4
)
8
⎤
⎥
⎦−
[
0 +
sin0
8
]
=
π
4
or 0. 7854
Problem 2. Determine
∫
sin^23 xdx.
Since cos2x= 1 −2sin^2 x(from Chapter 17),
then sin^2 x=
1
2
( 1 −cos 2x)and
sin^23 x =
1
2
( 1 −cos 6x)
Hence
∫
sin^23 xdx=
∫
1
2
( 1 −cos 6x)dx
=
1
2
(
x−
sin 6x
6
)
+c
Problem 3. Find 3
∫
tan^24 xdx.
Since 1+tan^2 x=sec^2 x,thentan^2 x=sec^2 x−1and
tan^24 x=sec^24 x−1.
Hence 3
∫
tan^24 xdx= 3
∫
(sec^24 x− 1 )dx
= 3
(
tan 4x
4
−x
)
+c