344 Higher Engineering Mathematics
5. (a) cosech −1 x
4
(b)
1
2
cosech
−1 4x
(a)
−4
x
(x 2 + 16)
(b)
−1
2x
(16x 2 + 1)
6. (a) coth −1 2x
7
(b)
1
4
coth −1 3t
(a)
14
49 − 4x 2 (b)
3
4(1 − 9t 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
(tanh x)
(a)
−1
(x − 1)
√
[x(2 − x)]
(b) 1
9. (a) cosh −1
t
t − 1
(b) coth −1 (cos x)
(a)
−1
(t − 1)
√
(2t − 1)
(b) −cosec x
10. (a) θ sinh −1 θ (b)
√
x cosh −1 x
⎡
⎢
⎢
⎢
⎣
(a)
θ
(θ 2 + 1)
+ sinh −1 θ
(b)
√
x
(x 2 − 1)
+
cosh
−1 x
2
√
x
⎤
⎥
⎥
⎥
⎦
11. (a)
2 sech
−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 + 2x tanh −1 x
(1 − x 2 ) 2
⎤
⎥
⎥
⎥
⎦
12. Show that
d
dx
[x cosh
−1
(cosh x)] = 2x.
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 2x
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 − 2x 2 )
dx
⎡
⎢
⎢
⎣
(a)
1
6
tan −1 θ
6
+ c
(b)
3
2
√
8
tanh −1 x
√
8
+ c
⎤
⎥
⎥
⎦