Higher Engineering Mathematics, Sixth Edition

344 Higher Engineering Mathematics

5. (a) cosech−^1

x

4

(b)

1

2

cosech−^14 x

[

(a)

− 4

x

√

(x^2 + 16 )

(b)

− 1

2 x

√

( 16 x^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 )

]

8. (a) sech−^1 (x− 1 )(b) tanh−^1 (tanhx)
   [
   (a)

− 1

(x− 1 )

√

[x( 2 −x)]

(b) 1

]

9. (a) cosh−^1

(

t

t− 1

)

(b) coth−^1 (cosx)

[

(a)

− 1

(t− 1 )

√

( 2 t− 1 )

(b)−cosecx

]

10. (a)θsinh−^1 θ(b)

√

xcosh−^1 x

⎡

⎢

⎢

⎢

⎣

(a)

θ

√

(θ^2 + 1 )

+sinh−^1 θ

(b)

√

x

√

(x^2 − 1 )

+

cosh−^1 x

2

√

x

⎤

⎥

⎥

⎥

⎦

11. (a)

2sech−^1

√

t

t^2

(b)

tanh−^1 x

( 1 −x^2 )

⎡

⎢

⎢

⎢

⎣

(a)

− 1

t^3

{

1

√

( 1 −t)

+4sech−^1

√

t

}

(b)

1 + 2 xtanh−^1 x

( 1 −x^2 )^2

⎤

⎥

⎥

⎥

⎦

12. Show that

d

dx

[xcosh−^1 (coshx)]= 2 x.

In Problems 13 to 15, determine the given

integrals.

13. (a)

∫

1

√

(x^2 + 9 )

dx

(b)

∫

3

√

( 4 x^2 + 25 )

dx

[

(a) sinh−^1

x

3

+c(b)

3

2

sinh−^1

2 x

5

+c

]

14. (a)

∫

1

√

(x^2 − 16 )

dx

(b)

∫

1

√

(t^2 − 5 )

dt

[

(a) cosh−^1

x

4

+c(b) cosh−^1

t

√

5

+c

]

15. (a)

∫

dθ

√

( 36 +θ^2 )

(b)

∫

3

( 16 − 2 x^2 )

dx

⎡

⎢

⎢

⎣

(a)

1

6

tan−^1

θ

6

+c

(b)

3

2

√

8

tanh−^1

x

√

8

+c

⎤

⎥

⎥

⎦