Higher Engineering Mathematics

INTEGRATION USING TRIGONOMETRIC AND HYPERBOLIC SUBSTITUTIONS 405

H

=

∫ √

(a^2 cosh^2 θ)(acoshθdθ),

since cosh^2 θ−sinh^2 θ= 1

=

∫

(acoshθ)(acoshθ)dθ=a^2

∫

cosh^2 θdθ

=a^2

∫ (

1 +cosh 2θ

2

)

dθ

=

a^2

2

(

θ+

sinh 2θ

2

)

+c

=

a^2

2

[θ+sinhθcoshθ]+c,

since sinh 2θ=2 sinhθcoshθ

Sincex=asinhθ, then sinhθ=

x

a

andθ=sinh−^1

x

a

Also since cosh^2 θ−sinh^2 θ= 1

then coshθ=

√

(1+sinh^2 θ)

=

√[

1 +

(x

a

) 2 ]

=

√(

a^2 +x^2

a^2

)

=

√

(a^2 +x^2 )

a

Hence

∫ √

(x^2 +a^2 )dx

=

a^2

2

[

sinh−^1

x

a

+

(x

a

)

√

(x^2 +a^2 )

a

]

+c

=

a^2

2

sinh−^1

x

a

+

x

2

√

(x^2 +a^2 )+c

Now try the following exercise.

Exercise 161 Further problems on integra-

tion using the sinhθsubstitution

1. Find

∫

2

√

(x^2 +16)

dx

[

2 sinh−^1

x

4

+c

]

2. Find

∫

3

√

(9+ 5 x^2 )

dx

[

3

√

5

sinh−^1

√

5

3

x+c

]

3. Find

∫ √

(x^2 +9) dx

[

9

2

sinh−^1

x

3

+

x

2

√

(x^2 +9)+c

]

4. Find

∫ √

(4t^2 +25) dt

[

25

4

sinh−^1

2 t

5

+

t

2

√

(4t^2 +25)+c

]

5. Evaluate

∫ 3

0

4

√

(t^2 +9)

dt [3.525]

6. Evaluate

∫ 1

0

√

(16+ 9 θ^2 )dθ [4.348]

40.8 Worked problems on integration

using the coshθsubstitution

Problem 24. Determine

∫

1

√

(x^2 −a^2 )

dx.

Letx=acoshθthen

dx

dθ

=asinhθand

dx=asinhθdθ

Hence

∫

1

√

(x^2 −a^2 )

dx

=

∫

1

√

(a^2 cosh^2 θ−a^2 )

(asinhθdθ)

=

∫

asinhθdθ

√

[a^2 ( cosh^2 θ−1)]

=

∫

asinhθdθ

√

(a^2 sinh^2 θ)

,

since cosh^2 θ−sinh^2 θ= 1

=

∫

asinhθdθ

asinhθ

=

∫

dθ=θ+c

=cosh−^1

x

a

+c, sincex=acoshθ

It is shown on page 337 that

cosh−^1

x

a

=ln

{

x+

√

(x^2 −a^2 )

a

}