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