The Laplace transform of e^3t is equal to 1/(s-3) and the Laplace of e^-3t is equal to 1/(s+3). This is because, we know that the Laplace of e^{at} is 1/(s-a).

The Laplace transform formula for the functions e^{3t} and e^{-3t} are given as follows.

- L{e
^{3t}} = $\dfrac{1}{s-3}$. - L{e
^{-3t}} = $\dfrac{1}{s+3}$.

## Laplace of e^{3t}

We will find the Laplace transform of e^{3t} by definition. The definition says that the Laplace of a function f(t) is given by the integral

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

Thus, the Laplace of e^{3t} by definition will be

L{e^{3t}} = $\int_0^\infty$ e^{-st} e^{3t} dt

= $\int_0^\infty$ e^{-(s-3)t} dt

= $\Big[ \dfrac{e^{-(s-3)t}}{-(s-3)}\Big]_0^\infty$

= lim_{t→∞} $\Big[ \dfrac{e^{-(s-3)t}}{-(s-3)}\Big]$ $-\dfrac{e^{0}}{-(s-3)}$

= 0 + $\dfrac{1}{s-3}$ as we know lim_{t→∞} e^{-(s-3)t} = 0 if s>3.

= $\dfrac{1}{s-3}$.

So the Laplace transform of e^{3t} is equal to 1/(s-3) when s>3, and this is proved by the definition of Laplace transforms.

L{e^{3t}} = 1/(s-3) whenever s>3 |

## Laplace of e^{-3t}

Replacing 3 by -3 in the above method of finding the Laplace transform of e^{3t}, we get that the Laplace transform of e^{-3t} is equal to 1/(s+3) if s> -3. That is,

L{e^{-3t}} = 1/(s+3) |

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## FAQs

### Q1: What is the Laplace transform of e^{3t}?

Answer: The Laplace transform of e^{3t} is equal to 1/(s-3) whenever s>3.