Laplace Transform of sin ax + cos ax
shahbaz ahmed
June 2024
Introduction to Laplace Transform
Let f(x) be defined for 0 ≤ x < ∞ and let s denote an arbitrary real
variable. The Laplace transform of f (x), designated by either L or
F (s), is
R∞
L {f (x)}=F (s)= 0 e−sx f (x)dx
for all values of s for which the improper integral converges.
Convergence occur when the limit
RR
limR−→∞ 0 e−sx f (x)dx
exists.
1 Formulas used
Prove that
If L {f (x)} = F (s), L {g(x)} = G(s), then for any two constants
c1 and c2
L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)}
Proof
L {c1 f (x) + c2 g(x)} =
R∞
e−sx [c1 f (x) + c2 g(x)] dx
R∞
R∞
L {c1 f (x) + c2 g(x)} = 0 e−sx [c1 f (x)] dx + 0 e−sx [c2 g(x)] dx
R∞
R∞
L {c1 f (x) + c2 g(x)} = c1 0 e−sx f (x) dx + c2 0 e−sx g(x) dx
0
L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)}
Also L {sin ax} =
L {cos ax} =
s
s2 +a2
a
s2 +a2
(s > 0)
(s > 0)
Find L {sin ax + cos ax}
Since
L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)}
Put c1 = c2 = 1
L {f (x) + g(x)} = L {f (x)} + L {g(x)}
2 Putting
f (x) = sin ax, g(x) = cos ax
L {sin ax + cos ax} = L {sin ax} + L {cos ax}
Putting
L {sin ax} =
a
s2 +a2
(s > 0)
L {cos ax} =
s
s2 +a2
(s > 0)
L {sin ax + cos ax} =
a
s2 +a2
L {sin ax + cos ax} =
a+s
s2 +a2
+
s
s2 +a2
(s > 0)
Applications
Find area under the curve
y = e−4x (sin 5x + cos 5x)
Solution
Here
s = 4, a = 5
Putting in the formula
L {sin ax + cos ax} =
Area =
4+5
42 +52
=
a+s
s2 +a2
(s > 0)
9
41
3
Laplace Transform of sin(ax)+cos(ax)
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