Higher Engineering Mathematics

342 DIFFERENTIAL CALCULUS

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)

√

(2t−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)

2 sec h−^1

√

t

t^2

(b)

tan h−^1 x

(1−x^2 )

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

(a)

− 1

t^3

{

1

√

(1−t)

+4 sech−^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

√

(4x^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

⎤

⎥

⎥

⎦