In mathematics, the Laplace transform is an integral transform that converts a function of time to a function of complex frequency. It is widely used in various fields of science and engineering to solve differential equations and determine system behavior.



The Laplace transform of e^-4t can be found using the following steps:

1. Define the function f(t) = e^-4t.

2. Apply the definition of the Laplace transform:

F(s) = L[f(t)] = ∫[0,∞] e^-st e^-4t dt

3. Simplify the integral by combining the exponents:

F(s) = ∫[0,∞] e^-(s+4)t dt

4. Use the formula for the Laplace transform of e^at:

L[e^at] = 1/(s-a)

5. Apply the formula to the simplified integral:

F(s) = 1/(s+4) ∫[0,∞] e^-(s+4)t dt

6. Evaluate the integral:

F(s) = 1/(s+4) [-1/(s+4) e^-(s+4)t] [0,∞]

7. Substitute the limits of integration:

F(s) = 1/(s+4) [-1/(s+4) (0 - 1)]

F(s) = 1/(s+4)^2



Therefore, the Laplace transform of e^-4t is 1/(s+4)^2. This result can be used to solve differential equations involving the function e^-4t.