Products of Trigonometric Functions and Ex-
ponentials
(117)
Z
e
x
sin x dx =
1
2
e
x
(sin x - cos x)
(118)
Z
e
bx
sin ax dx =
1
a
2
+ b
2
e
bx
(b sin ax - a cos ax)
(119)
Z
e
x
cos x dx =
1
2
e
x
(sin x + cos x)
(120)
Z
e
bx
cos ax dx =
1
a
2
+ b
2
e
bx
(a sin ax + b cos ax)
(121)
Z
xe
x
sin x dx =
1
2
e
x
(cos x - x cos x + x sin x)
(122)
Z
xe
x
cos x dx =
1
2
e
x
(x cos x - sin x + x sin x)
Integrals of Hyperbolic Functions
(123)
Z
cosh ax dx =
1
a
sinh ax
(124)
Z
e
ax
cosh bx dx =
e
ax
a
2
- b
2
[a cosh bx - b sinh bx] a 6= b
e
2ax
4a
+
x
2
a = b
(125)
Z
sinh ax dx =
1
a
cosh ax
13
(126)
Z
e
ax
sinh bx dx =
e
ax
a
2
- b
2
[-b cosh bx + a sinh bx] a 6= b
e
2ax
4a
-
x
2
a = b
(127)
Z
tanh ax dx =
1
a
ln cosh ax
(128)
Z
e
ax
tanh bx dx =
e
(a+2b)x
(a +2b)
2
F
1
h
1+
a
2b
, 1, 2+
a
2b
, -e
2bx
i
-
1
a
e
ax
2
F
1
h
1,
a
2b
, 1+
a
2b
, -e
2bx
i
a 6= b
e
ax
- 2 tan
-1
[e
ax
]
a
a = b
(129)
Z
cos ax cosh bx dx =
1
a
2
+ b
2
[a sin ax cosh bx + b cos ax sinh bx]
(130)
Z
cos ax sinh bx dx =
1
a
2
+ b
2
[b cos ax cosh bx + a sin ax sinh bx]
(131)
Z
sin ax cosh bx dx =
1
a
2
+ b
2
[-a cos ax cosh bx + b sin ax sinh bx]
(132)
Z
sin ax sinh bx dx =
1
a
2
+ b
2
[b cosh bx sin ax - a cos ax sinh bx]
(133)
Z
sinh ax cosh axdx =
1
4a
[-2ax + sinh 2ax]
(134)
Z
sinh ax cosh bx dx =
1
b
2
- a
2
[b cosh bx sinh ax - a cosh ax sinh bx]
c 2014. From http://integral-table.com, last revised June 14, 2014. This mate-
rial is provided as is without warranty or representation about the accuracy, correctness or
suitability of this material for any purpose. This work is licensed under the Creative Com-
mons Attribution-Noncommercial-Share Alike 3.0 United States License. To view a copy
of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send
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