Higher Engineering Mathematics

DIFFERENTIATION OF INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 341

G

Problem 19. Determine

∫

dx

√

(x^2 +4)

Since

d

dx

(

sinh−^1

x

a

)

=

1

√

(x^2 +a^2 )

then

∫

dx

√

(x^2 +a^2 )

=sinh−^1

x

a

+c

Hence

∫

1

√

(x^2 +4)

dx =

∫

1

√

(x^2 + 22 )

dx

=sinh−^1

x

2

+c

Problem 20. Determine

∫

4

√

(x^2 −3)

dx.

Since

d

dx

(

cosh−^1

x

a

)

=

1

√

(x^2 −a^2 )

then

∫

1

√

(x^2 −a^2 )

dx=cosh−^1

x

a

+c

Hence

∫

4

√

(x^2 −3)

dx= 4

∫

1

√

[x^2 −(

√

3)^2 ]

dx

=4 cosh−^1

x

√

3

+c

Problem 21. Find

∫

2

(9− 4 x^2 )

dx.

Since tanh−^1

x

a

=

a

a^2 −x^2

then

∫

a

a^2 −x^2

dx=tanh−^1

x

a

+c

i.e.

∫

1

a^2 −x^2

dx=

1

a

tanh−^1

x

a

+c

Hence

∫

2

(9− 4 x^2 )

dx= 2

∫

1

4

( 9

4 −x

2

)dx

=

1

2

∫

1

[(

3

2

) 2

−x^2

]dx

=

1

2

[

1

( 3

2

)tanh−^1

x

( 3

2

)+c

]

i.e.

∫

2

(9− 4 x^2 )

dx=

1

3

tanh−^1

2 x

3

+c

Now try the following exercise.

Exercise 138 Further problems on differen-

tiation of inverse hyperbolic functions

In Problems 1 to 11, differentiate with respect to

the variable.

1. (a) sinh−^1

x

3

(b) sinh−^14 x

[

(a)

1

√

(x^2 +9)

(b)

4

√

(16x^2 +1)

]

2. (a) 2 cosh−^1

t

3

(b)

1

2

cosh−^12 θ

[

(a)

2

√

(t^2 −9)

(b)

1

√

(4θ^2 −1)

]

3. (a) tanh−^1

2 x

5

(b) 3 tanh−^13 x

[

(a)

10

25 − 4 x^2

(b)

9

(1− 9 x^2 )

]

4. (a) sech−^1

3 x

4

(b)−

1

2

sech−^12 x

[

(a)

− 4

x

√

(16− 9 x^2 )

(b)

1

2 x

√

(1− 4 x^2 )

]

5. (a) cosech−^1

x

4

(b)

1

2

cosech−^14 x

[

(a)

− 4

x

√

(x^2 +16)

(b)

− 1

2 x

√

(16x^2 +1)

]

6. (a) coth−^1

2 x

7

(b)

1

4

coth−^13 t

[

(a)

14

49 − 4 x^2

(b)

3

4(1− 9 t^2 )

]

7. (a) 2 sinh−^1

√

(x^2 −1)

(b)

1

2

cosh−^1

√

(x^2 +1)

[

(a)

2

√

(x^2 −1)

(b)

1

2

√

(x^2 +1)

]