Find Laplace Transform of e^4t | Laplace of e^-4t

The Laplace transform of e^4t is equal to 1/(s-4) and the Laplace of e^-4t is equal to 1/(s+4). This is because, we know that the Laplace of e^at is 1/(s-a).

The Laplace transform formulae for the functions e^4t and e^-4t are given in the table below.

Function f(t)	L{ f(t) }
e4t	L{e4t} = $\dfrac{1}{s-4}$
e-4t	L{e-4t} = $\dfrac{1}{s+4}$

Laplace of e^4t

Let us find the Laplace transform of e^4t by definition. The Laplace transform of a function f(t) by definition is given by the following integral formula

L{f(t)} = $\int_0^\infty$ e^-st f(t) dt.

Thus, the Laplace of e^4t by definition will be

L{e^4t} = $\int_0^\infty$ e^-st e^4t dt

= $\int_0^\infty$ e^-(s-4)t dt

= $\Big[ \dfrac{e^{-(s-4)t}}{-(s-4)}\Big]_0^\infty$

= lim_t→∞ $\Big[ \dfrac{e^{-(s-4)t}}{-(s-4)}\Big]$ $-\dfrac{e^{0}}{-(s-4)}$

= 0 + $\dfrac{1}{s-4}$ as we know lim_t→∞ e^-(s-4)t = 0 when s>4.

= $\dfrac{1}{s-4}$.

So the Laplace transform of e^4t is equal to 1/(s-4) if s>4, and this is obtained by the definition of Laplace transforms.

L{e4t} = 1/(s-4) whenever s>4

Laplace of e^-4t

Replacing 4 by -4 in the above method of finding the Laplace transform of e^4t, we get the Laplace transform of e^-4t which is equal to 1/(s+4) if s> -4. That is,

L{e-4t} = 1/(s+4)

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