
27.1. Indefinite Integrals 1445
14.
Z
dx
sinh
2n
x
=
cosh x
2n − 1
−
1
sinh
2n−1
x
+
n−1
X
k=1
(−1)
k−1
A
nk
sinh
2n−2k −1
x
,
A
nk
=
2
k
(n − 1)(n −2) . . . (n − k)
(2n − 3)(2n −5) . . . (2n − 2k − 1)
n = 1, 2, . . .
15.
Z
dx
sinh
2n+1
x
=
cosh x
2n
−
1
sinh
2n
x
+
n−1
X
k=1
(−1)
k−1
A
nk
sinh
2n−2k
x
+ (−1)
n
(2n − 1)!!
(2n)!!
ln tanh
x
2
, A
nk
=
(2n − 1)(2n −3) . . . (2n − 2k + 1)
2
k
(n − 1)(n −2) . . . (n − k)
,
n = 1, 2, . . .
16.
Z
dx
a + b sinh x
=
1
√
a
2
+ b
2
ln
a tanh(x/2) −b +
√
a
2
+ b
2
a tanh(x/2) −b −
√
a
2
+ b
2
.
17.
Z
Ax + B sinh x
a + b sinh x
dx =
B
b
x +
Ab −Ba
b
√
a
2
+ b
2
ln
a tanh(x/2) − b +
√
a
2
+ b
2
a tanh(x/2) − b −
√
a
2
+ b
2
.
◮ Integrals involving tanhx or coth x.
1.
Z
tanh x dx = ln cosh x.
2.
Z
tanh
2
x dx = x − tanh x.
3.
Z
tanh
3
x dx =