Laplace and fourier transform of one sided decaying exponential

Question

I am stuck on this problem as i have no idea what will be the limits.. and also don't know how to start from till the end.

1 This is question: 1

Answer 1

The Laplace Transform of $f(t)=e^{-at}u(t)$ is given by

$$\mathscr{L}\{e^{-at}u(t)\}=\int_0^\infty e^{-at}e^{-st}\,dt=\frac{1}{s+a}$$

The Fourier Transform of $f(t)$ is given by

$$\mathscr{F}\{e^{-at}u(t)\}=\int_0^\infty e^{-at}e^{i\omega t}\,dt=\frac{1}{a-i\omega}$$