
1468 INTEGRAL TRANSFORMS
No. Original function, f(x)
Laplace transform,
e
f(p) =
Z
∞
0
e
−px
f(x) dx
8
1
x
2
sin
2
(ax)
a arctan(2a/p) −
1
4
p ln
1 + 4a
2
p
−2
9 sin
2
√
ax
√
πa
p
√
p
e
−a/p
10
1
x
sin
2
√
ax
π erf
p
a/p
11 cos(ax)
p
p
2
+ a
2
12
cos
2
(ax)
p
2
+ 2a
2
p
p
2
+ 4a
2
13 x
n
cos(ax), n = 1, 2, . . .
n! p
n+1
p
2
+ a
2
n+1
X
0≤2k≤n+1
(−1)
k
C
2k
n+1
a
p
2k
14
1
x
1 − cos(ax)
1
2
ln
1 + a
2
p
−2
15
1
x
cos(ax) − cos(bx)
1
2
ln
p
2
+ b
2
p
2
+ a
2
16
√
x cos
2
√
ax
1
2
π
1/2
p
−5/2
(p − 2a)e
−a/p
17
1
√
x
cos
2
√
ax
p
π/p
e
−a/p
18 sin(ax) sin(bx)
2abp
p
2
+ (a + b)
2
p
2
+ (a − b)
2
19 cos(ax) sin(bx)
b
p
2
− a
2
+ b
2
p
2
+ (a + b)
2
p
2
+ (a − b)
2
20 cos(ax) cos(bx)
p
p
2
+ a
2
+ b
2
p
2
+ (a + b)
2
p
2
+ (a − b)
2
21
ax cos(ax) −sin (ax)
x
2
p arctan
a
x
− a
22
e
bx
sin(ax